Theory of topology on ordered spaces #
Main definitions #
The order topology on an ordered space is the topology generated by all open intervals (or
equivalently by those of the form (-∞, a) and (b, +∞)). We define it as preorder.topology α.
However, we do not register it as an instance (as many existing ordered types already have
topologies, which would be equal but not definitionally equal to preorder.topology α). Instead,
we introduce a class order_topology α(which is a Prop, also known as a mixin) saying that on
the type α having already a topological space structure and a preorder structure, the topological
structure is equal to the order topology.
We also introduce another (mixin) class order_closed_topology α saying that the set of points
(x, y) with x ≤ y is closed in the product space. This is automatically satisfied on a linear
order with the order topology.
We prove many basic properties of such topologies.
Main statements #
This file contains the proofs of the following facts. For exact requirements
(order_closed_topology vs order_topology, preorder vs partial_order vs linear_order etc)
see their statements.
Open / closed sets #
is_open_lt: iffandgare continuous functions, then{x | f x < g x}is open;is_open_Iio,is_open_Ioi,is_open_Ioo: open intervals are open;is_closed_le: iffandgare continuous functions, then{x | f x ≤ g x}is closed;is_closed_Iic,is_closed_Ici,is_closed_Icc: closed intervals are closed;frontier_le_subset_eq,frontier_lt_subset_eq: frontiers of both{x | f x ≤ g x}and{x | f x < g x}are included by{x | f x = g x};exists_Ioc_subset_of_mem_nhds,exists_Ico_subset_of_mem_nhds: ifx < y, then any neighborhood ofxincludes an interval[x, z)for somez ∈ (x, y], and any neighborhood ofyincludes an interval(z, y]for somez ∈ [x, y).
Convergence and inequalities #
le_of_tendsto_of_tendsto: iffconverges toa,gconverges tob, and eventuallyf x ≤ g x, thena ≤ ble_of_tendsto,ge_of_tendsto: iffconverges toaand eventuallyf x ≤ b(resp.,b ≤ f x), thena ≤ b(resp.,b ≤ a); we also provide primed versions that assume the inequalities to hold for allx`.
Min, max, Sup and Inf #
continuous.min,continuous.max: pointwisemin/maxof two continuous functions is continuous.tendsto.min,tendsto.max: ifftends toaandgtends tob, then their pointwisemin/maxtend tomin a bandmax a b, respectively.tendsto_of_tendsto_of_tendsto_of_le_of_le: theorem known as squeeze theorem, sandwich theorem, theorem of Carabinieri, and two policemen (and a drunk) theorem; ifgandhboth converge toa, and eventuallyg x ≤ f x ≤ h x, thenfconverges toa.
Connected sets and Intermediate Value Theorem #
is_preconnected_I??: all intervalsI??are preconnected,is_preconnected.intermediate_value,intermediate_value_univ: Intermediate Value Theorem for connected sets and connected spaces, respectively;intermediate_value_Icc,intermediate_value_Icc': Intermediate Value Theorem for functions on closed intervals.
Miscellaneous facts #
is_compact.exists_forall_le,is_compact.exists_forall_ge: extreme value theorem, a continuous function on a compact set takes its minimum and maximum values.is_closed.Icc_subset_of_forall_mem_nhds_within: “Continuous induction” principle; ifs ∩ [a, b]is closed,a ∈ s, and for eachx ∈ [a, b) ∩ ssome of its right neighborhoods is includeds, then[a, b] ⊆ s.is_closed.Icc_subset_of_forall_exists_gt,is_closed.mem_of_ge_of_forall_exists_gt: two other versions of the “continuous induction” principle.
Implementation #
We do not register the order topology as an instance on a preorder (or even on a linear order).
Indeed, on many such spaces, a topology has already been constructed in a different way (think
of the discrete spaces ℕ or ℤ, or ℝ that could inherit a topology as the completion of ℚ),
and is in general not defeq to the one generated by the intervals. We make it available as a
definition preorder.topology α though, that can be registered as an instance when necessary, or
for specific types.
@[class]
A topology on a set which is both a topological space and a preorder is order-closed if the
set of points (x, y) with x ≤ y is closed in the product space. We introduce this as a mixin.
This property is satisfied for the order topology on a linear order, but it can be satisfied more
generally, and suffices to derive many interesting properties relating order and topology.
@[protected, instance]
Equations
theorem
is_closed_le_prod
{α : Type u}
[topological_space α]
[preorder α]
[t : order_closed_topology α] :
theorem
is_closed_le
{α : Type u}
{β : Type v}
[topological_space α]
[preorder α]
[t : order_closed_topology α]
[topological_space β]
{f g : β → α}
(hf : continuous f)
(hg : continuous g) :
theorem
is_closed_le'
{α : Type u}
[topological_space α]
[preorder α]
[t : order_closed_topology α]
(a : α) :
theorem
is_closed_Iic
{α : Type u}
[topological_space α]
[preorder α]
[t : order_closed_topology α]
{a : α} :
theorem
is_closed_ge'
{α : Type u}
[topological_space α]
[preorder α]
[t : order_closed_topology α]
(a : α) :
theorem
is_closed_Ici
{α : Type u}
[topological_space α]
[preorder α]
[t : order_closed_topology α]
{a : α} :
@[protected, instance]
def
order_dual.order_closed_topology
{α : Type u}
[topological_space α]
[preorder α]
[t : order_closed_topology α] :
theorem
is_closed_Icc
{α : Type u}
[topological_space α]
[preorder α]
[t : order_closed_topology α]
{a b : α} :
@[simp]
theorem
closure_Icc
{α : Type u}
[topological_space α]
[preorder α]
[t : order_closed_topology α]
(a b : α) :
@[simp]
theorem
closure_Iic
{α : Type u}
[topological_space α]
[preorder α]
[t : order_closed_topology α]
(a : α) :
@[simp]
theorem
closure_Ici
{α : Type u}
[topological_space α]
[preorder α]
[t : order_closed_topology α]
(a : α) :
theorem
le_of_tendsto_of_tendsto
{α : Type u}
{β : Type v}
[topological_space α]
[preorder α]
[t : order_closed_topology α]
{f g : β → α}
{b : filter β}
{a₁ a₂ : α}
[b.ne_bot]
(hf : filter.tendsto f b (𝓝 a₁))
(hg : filter.tendsto g b (𝓝 a₂))
(h : f ≤ᶠ[b] g) :
a₁ ≤ a₂
theorem
le_of_tendsto_of_tendsto'
{α : Type u}
{β : Type v}
[topological_space α]
[preorder α]
[t : order_closed_topology α]
{f g : β → α}
{b : filter β}
{a₁ a₂ : α}
[b.ne_bot]
(hf : filter.tendsto f b (𝓝 a₁))
(hg : filter.tendsto g b (𝓝 a₂))
(h : ∀ (x : β), f x ≤ g x) :
a₁ ≤ a₂
theorem
le_of_tendsto
{α : Type u}
{β : Type v}
[topological_space α]
[preorder α]
[t : order_closed_topology α]
{f : β → α}
{a b : α}
{x : filter β}
[x.ne_bot]
(lim : filter.tendsto f x (𝓝 a))
(h : ∀ᶠ (c : β) in x, f c ≤ b) :
a ≤ b
theorem
le_of_tendsto'
{α : Type u}
{β : Type v}
[topological_space α]
[preorder α]
[t : order_closed_topology α]
{f : β → α}
{a b : α}
{x : filter β}
[x.ne_bot]
(lim : filter.tendsto f x (𝓝 a))
(h : ∀ (c : β), f c ≤ b) :
a ≤ b
theorem
ge_of_tendsto
{α : Type u}
{β : Type v}
[topological_space α]
[preorder α]
[t : order_closed_topology α]
{f : β → α}
{a b : α}
{x : filter β}
[x.ne_bot]
(lim : filter.tendsto f x (𝓝 a))
(h : ∀ᶠ (c : β) in x, b ≤ f c) :
b ≤ a
theorem
ge_of_tendsto'
{α : Type u}
{β : Type v}
[topological_space α]
[preorder α]
[t : order_closed_topology α]
{f : β → α}
{a b : α}
{x : filter β}
[x.ne_bot]
(lim : filter.tendsto f x (𝓝 a))
(h : ∀ (c : β), b ≤ f c) :
b ≤ a
@[simp]
theorem
closure_le_eq
{α : Type u}
{β : Type v}
[topological_space α]
[preorder α]
[t : order_closed_topology α]
[topological_space β]
{f g : β → α}
(hf : continuous f)
(hg : continuous g) :
theorem
closure_lt_subset_le
{α : Type u}
{β : Type v}
[topological_space α]
[preorder α]
[t : order_closed_topology α]
[topological_space β]
{f g : β → α}
(hf : continuous f)
(hg : continuous g) :
theorem
continuous_within_at.closure_le
{α : Type u}
{β : Type v}
[topological_space α]
[preorder α]
[t : order_closed_topology α]
[topological_space β]
{f g : β → α}
{s : set β}
{x : β}
(hx : x ∈ closure s)
(hf : continuous_within_at f s x)
(hg : continuous_within_at g s x)
(h : ∀ (y : β), y ∈ s → f y ≤ g y) :
f x ≤ g x
theorem
is_closed.is_closed_le
{α : Type u}
{β : Type v}
[topological_space α]
[preorder α]
[t : order_closed_topology α]
[topological_space β]
{f g : β → α}
{s : set β}
(hs : is_closed s)
(hf : continuous_on f s)
(hg : continuous_on g s) :
If s is a closed set and two functions f and g are continuous on s,
then the set {x ∈ s | f x ≤ g x} is a closed set.
theorem
nhds_within_Ici_ne_bot
{α : Type u}
[topological_space α]
[preorder α]
{a b : α}
(H₂ : a ≤ b) :
@[instance]
theorem
nhds_within_Iic_ne_bot
{α : Type u}
[topological_space α]
[preorder α]
{a b : α}
(H : a ≤ b) :
@[instance]
@[protected, instance]
def
order_closed_topology.to_t2_space
{α : Type u}
[topological_space α]
[partial_order α]
[t : order_closed_topology α] :
t2_space α
theorem
is_open_lt_prod
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α] :
theorem
is_open_lt
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[order_closed_topology α]
[topological_space β]
{f g : β → α}
(hf : continuous f)
(hg : continuous g) :
theorem
is_open_Iio
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a : α} :
theorem
is_open_Ioi
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a : α} :
theorem
is_open_Ioo
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a b : α} :
@[simp]
theorem
interior_Ioi
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a : α} :
@[simp]
theorem
interior_Iio
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a : α} :
@[simp]
theorem
interior_Ioo
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a b : α} :
theorem
eventually_le_of_tendsto_lt
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{l : filter γ}
{f : γ → α}
{u v : α}
(hv : v < u)
(h : filter.tendsto f l (𝓝 v)) :
∀ᶠ (a : γ) in l, f a ≤ u
theorem
eventually_ge_of_tendsto_gt
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{l : filter γ}
{f : γ → α}
{u v : α}
(hv : u < v)
(h : filter.tendsto f l (𝓝 v)) :
∀ᶠ (a : γ) in l, u ≤ f a
theorem
intermediate_value_univ₂
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_closed_topology α]
[topological_space γ]
[preconnected_space γ]
{a b : γ}
{f g : γ → α}
(hf : continuous f)
(hg : continuous g)
(ha : f a ≤ g a)
(hb : g b ≤ f b) :
∃ (x : γ), f x = g x
Intermediate value theorem for two functions: if f and g are two continuous functions
on a preconnected space and f a ≤ g a and g b ≤ f b, then for some x we have f x = g x.
theorem
intermediate_value_univ₂_eventually₁
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_closed_topology α]
[topological_space γ]
[preconnected_space γ]
{a : γ}
{l : filter γ}
[l.ne_bot]
{f g : γ → α}
(hf : continuous f)
(hg : continuous g)
(ha : f a ≤ g a)
(he : g ≤ᶠ[l] f) :
∃ (x : γ), f x = g x
theorem
intermediate_value_univ₂_eventually₂
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_closed_topology α]
[topological_space γ]
[preconnected_space γ]
{l₁ l₂ : filter γ}
[l₁.ne_bot]
[l₂.ne_bot]
{f g : γ → α}
(hf : continuous f)
(hg : continuous g)
(he₁ : f ≤ᶠ[l₁] g)
(he₂ : g ≤ᶠ[l₂] f) :
∃ (x : γ), f x = g x
theorem
is_preconnected.intermediate_value₂
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_closed_topology α]
[topological_space γ]
{s : set γ}
(hs : is_preconnected s)
{a b : γ}
(ha : a ∈ s)
(hb : b ∈ s)
{f g : γ → α}
(hf : continuous_on f s)
(hg : continuous_on g s)
(ha' : f a ≤ g a)
(hb' : g b ≤ f b) :
Intermediate value theorem for two functions: if f and g are two functions continuous
on a preconnected set s and for some a b ∈ s we have f a ≤ g a and g b ≤ f b,
then for some x ∈ s we have f x = g x.
theorem
is_preconnected.intermediate_value₂_eventually₁
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_closed_topology α]
[topological_space γ]
{s : set γ}
(hs : is_preconnected s)
{a : γ}
{l : filter γ}
(ha : a ∈ s)
[l.ne_bot]
(hl : l ≤ 𝓟 s)
{f g : γ → α}
(hf : continuous_on f s)
(hg : continuous_on g s)
(ha' : f a ≤ g a)
(he : g ≤ᶠ[l] f) :
theorem
is_preconnected.intermediate_value₂_eventually₂
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_closed_topology α]
[topological_space γ]
{s : set γ}
(hs : is_preconnected s)
{l₁ l₂ : filter γ}
[l₁.ne_bot]
[l₂.ne_bot]
(hl₁ : l₁ ≤ 𝓟 s)
(hl₂ : l₂ ≤ 𝓟 s)
{f g : γ → α}
(hf : continuous_on f s)
(hg : continuous_on g s)
(he₁ : f ≤ᶠ[l₁] g)
(he₂ : g ≤ᶠ[l₂] f) :
theorem
is_preconnected.intermediate_value
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_closed_topology α]
[topological_space γ]
{s : set γ}
(hs : is_preconnected s)
{a b : γ}
(ha : a ∈ s)
(hb : b ∈ s)
{f : γ → α}
(hf : continuous_on f s) :
Intermediate Value Theorem for continuous functions on connected sets.
theorem
is_preconnected.intermediate_value_Ico
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_closed_topology α]
[topological_space γ]
{s : set γ}
(hs : is_preconnected s)
{a : γ}
{l : filter γ}
(ha : a ∈ s)
[l.ne_bot]
(hl : l ≤ 𝓟 s)
{f : γ → α}
(hf : continuous_on f s)
{v : α}
(ht : filter.tendsto f l (𝓝 v)) :
theorem
is_preconnected.intermediate_value_Ioc
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_closed_topology α]
[topological_space γ]
{s : set γ}
(hs : is_preconnected s)
{a : γ}
{l : filter γ}
(ha : a ∈ s)
[l.ne_bot]
(hl : l ≤ 𝓟 s)
{f : γ → α}
(hf : continuous_on f s)
{v : α}
(ht : filter.tendsto f l (𝓝 v)) :
theorem
is_preconnected.intermediate_value_Ioo
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_closed_topology α]
[topological_space γ]
{s : set γ}
(hs : is_preconnected s)
{l₁ l₂ : filter γ}
[l₁.ne_bot]
[l₂.ne_bot]
(hl₁ : l₁ ≤ 𝓟 s)
(hl₂ : l₂ ≤ 𝓟 s)
{f : γ → α}
(hf : continuous_on f s)
{v₁ v₂ : α}
(ht₁ : filter.tendsto f l₁ (𝓝 v₁))
(ht₂ : filter.tendsto f l₂ (𝓝 v₂)) :
theorem
is_preconnected.intermediate_value_Ici
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_closed_topology α]
[topological_space γ]
{s : set γ}
(hs : is_preconnected s)
{a : γ}
{l : filter γ}
(ha : a ∈ s)
[l.ne_bot]
(hl : l ≤ 𝓟 s)
{f : γ → α}
(hf : continuous_on f s)
(ht : filter.tendsto f l filter.at_top) :
theorem
is_preconnected.intermediate_value_Iic
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_closed_topology α]
[topological_space γ]
{s : set γ}
(hs : is_preconnected s)
{a : γ}
{l : filter γ}
(ha : a ∈ s)
[l.ne_bot]
(hl : l ≤ 𝓟 s)
{f : γ → α}
(hf : continuous_on f s)
(ht : filter.tendsto f l filter.at_bot) :
theorem
is_preconnected.intermediate_value_Ioi
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_closed_topology α]
[topological_space γ]
{s : set γ}
(hs : is_preconnected s)
{l₁ l₂ : filter γ}
[l₁.ne_bot]
[l₂.ne_bot]
(hl₁ : l₁ ≤ 𝓟 s)
(hl₂ : l₂ ≤ 𝓟 s)
{f : γ → α}
(hf : continuous_on f s)
{v : α}
(ht₁ : filter.tendsto f l₁ (𝓝 v))
(ht₂ : filter.tendsto f l₂ filter.at_top) :
theorem
is_preconnected.intermediate_value_Iio
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_closed_topology α]
[topological_space γ]
{s : set γ}
(hs : is_preconnected s)
{l₁ l₂ : filter γ}
[l₁.ne_bot]
[l₂.ne_bot]
(hl₁ : l₁ ≤ 𝓟 s)
(hl₂ : l₂ ≤ 𝓟 s)
{f : γ → α}
(hf : continuous_on f s)
{v : α}
(ht₁ : filter.tendsto f l₁ filter.at_bot)
(ht₂ : filter.tendsto f l₂ (𝓝 v)) :
theorem
is_preconnected.intermediate_value_Iii
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_closed_topology α]
[topological_space γ]
{s : set γ}
(hs : is_preconnected s)
{l₁ l₂ : filter γ}
[l₁.ne_bot]
[l₂.ne_bot]
(hl₁ : l₁ ≤ 𝓟 s)
(hl₂ : l₂ ≤ 𝓟 s)
{f : γ → α}
(hf : continuous_on f s)
(ht₁ : filter.tendsto f l₁ filter.at_bot)
(ht₂ : filter.tendsto f l₂ filter.at_top) :
theorem
intermediate_value_univ
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_closed_topology α]
[topological_space γ]
[preconnected_space γ]
(a b : γ)
{f : γ → α}
(hf : continuous f) :
Intermediate Value Theorem for continuous functions on connected spaces.
theorem
mem_range_of_exists_le_of_exists_ge
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_closed_topology α]
[topological_space γ]
[preconnected_space γ]
{c : α}
{f : γ → α}
(hf : continuous f)
(h₁ : ∃ (a : γ), f a ≤ c)
(h₂ : ∃ (b : γ), c ≤ f b) :
Intermediate Value Theorem for continuous functions on connected spaces.
theorem
is_preconnected.Icc_subset
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{s : set α}
(hs : is_preconnected s)
{a b : α}
(ha : a ∈ s)
(hb : b ∈ s) :
If a preconnected set contains endpoints of an interval, then it includes the whole interval.
theorem
is_connected.Icc_subset
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{s : set α}
(hs : is_connected s)
{a b : α}
(ha : a ∈ s)
(hb : b ∈ s) :
If a preconnected set contains endpoints of an interval, then it includes the whole interval.
theorem
is_preconnected.eq_univ_of_unbounded
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{s : set α}
(hs : is_preconnected s)
(hb : ¬bdd_below s)
(ha : ¬bdd_above s) :
If preconnected set in a linear order space is unbounded below and above, then it is the whole space.
Neighborhoods to the left and to the right on an order_closed_topology #
Limits to the left and to the right of real functions are defined in terms of neighborhoods to
the left and to the right, either open or closed, i.e., members of 𝓝[Ioi a] a and
𝓝[Ici a] a on the right, and similarly on the left. Here we simply prove that all
right-neighborhoods of a point are equal, and we'll prove later other useful characterizations which
require the stronger hypothesis order_topology α
Right neighborhoods, point excluded #
theorem
Ioo_mem_nhds_within_Ioi
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a b c : α}
(H : b ∈ set.Ico a c) :
theorem
Ioc_mem_nhds_within_Ioi
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a b c : α}
(H : b ∈ set.Ico a c) :
theorem
Ico_mem_nhds_within_Ioi
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a b c : α}
(H : b ∈ set.Ico a c) :
theorem
Icc_mem_nhds_within_Ioi
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a b c : α}
(H : b ∈ set.Ico a c) :
@[simp]
theorem
nhds_within_Ioc_eq_nhds_within_Ioi
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a b : α}
(h : a < b) :
@[simp]
theorem
nhds_within_Ioo_eq_nhds_within_Ioi
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a b : α}
(h : a < b) :
@[simp]
theorem
continuous_within_at_Ioc_iff_Ioi
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[order_closed_topology α]
[topological_space β]
{a b : α}
{f : α → β}
(h : a < b) :
continuous_within_at f (set.Ioc a b) a ↔ continuous_within_at f (set.Ioi a) a
@[simp]
theorem
continuous_within_at_Ioo_iff_Ioi
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[order_closed_topology α]
[topological_space β]
{a b : α}
{f : α → β}
(h : a < b) :
continuous_within_at f (set.Ioo a b) a ↔ continuous_within_at f (set.Ioi a) a
Left neighborhoods, point excluded #
theorem
Ioo_mem_nhds_within_Iio
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a b c : α}
(H : b ∈ set.Ioc a c) :
theorem
Ico_mem_nhds_within_Iio
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a b c : α}
(H : b ∈ set.Ioc a c) :
theorem
Ioc_mem_nhds_within_Iio
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a b c : α}
(H : b ∈ set.Ioc a c) :
theorem
Icc_mem_nhds_within_Iio
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a b c : α}
(H : b ∈ set.Ioc a c) :
@[simp]
theorem
nhds_within_Ico_eq_nhds_within_Iio
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a b : α}
(h : a < b) :
@[simp]
theorem
nhds_within_Ioo_eq_nhds_within_Iio
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a b : α}
(h : a < b) :
@[simp]
theorem
continuous_within_at_Ico_iff_Iio
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_closed_topology α]
[topological_space γ]
{a b : α}
{f : α → γ}
(h : a < b) :
continuous_within_at f (set.Ico a b) b ↔ continuous_within_at f (set.Iio b) b
@[simp]
theorem
continuous_within_at_Ioo_iff_Iio
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_closed_topology α]
[topological_space γ]
{a b : α}
{f : α → γ}
(h : a < b) :
continuous_within_at f (set.Ioo a b) b ↔ continuous_within_at f (set.Iio b) b
Right neighborhoods, point included #
theorem
Ioo_mem_nhds_within_Ici
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a b c : α}
(H : b ∈ set.Ioo a c) :
theorem
Ioc_mem_nhds_within_Ici
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a b c : α}
(H : b ∈ set.Ioo a c) :
theorem
Ico_mem_nhds_within_Ici
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a b c : α}
(H : b ∈ set.Ico a c) :
theorem
Icc_mem_nhds_within_Ici
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a b c : α}
(H : b ∈ set.Ico a c) :
@[simp]
theorem
nhds_within_Icc_eq_nhds_within_Ici
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a b : α}
(h : a < b) :
@[simp]
theorem
nhds_within_Ico_eq_nhds_within_Ici
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a b : α}
(h : a < b) :
@[simp]
theorem
continuous_within_at_Icc_iff_Ici
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[order_closed_topology α]
[topological_space β]
{a b : α}
{f : α → β}
(h : a < b) :
continuous_within_at f (set.Icc a b) a ↔ continuous_within_at f (set.Ici a) a
@[simp]
theorem
continuous_within_at_Ico_iff_Ici
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[order_closed_topology α]
[topological_space β]
{a b : α}
{f : α → β}
(h : a < b) :
continuous_within_at f (set.Ico a b) a ↔ continuous_within_at f (set.Ici a) a
Left neighborhoods, point included #
theorem
Ioo_mem_nhds_within_Iic
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a b c : α}
(H : b ∈ set.Ioo a c) :
theorem
Ico_mem_nhds_within_Iic
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a b c : α}
(H : b ∈ set.Ioo a c) :
theorem
Ioc_mem_nhds_within_Iic
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a b c : α}
(H : b ∈ set.Ioc a c) :
theorem
Icc_mem_nhds_within_Iic
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a b c : α}
(H : b ∈ set.Ioc a c) :
@[simp]
theorem
nhds_within_Icc_eq_nhds_within_Iic
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a b : α}
(h : a < b) :
@[simp]
theorem
nhds_within_Ioc_eq_nhds_within_Iic
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{a b : α}
(h : a < b) :
@[simp]
theorem
continuous_within_at_Icc_iff_Iic
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[order_closed_topology α]
[topological_space β]
{a b : α}
{f : α → β}
(h : a < b) :
continuous_within_at f (set.Icc a b) b ↔ continuous_within_at f (set.Iic b) b
@[simp]
theorem
continuous_within_at_Ioc_iff_Iic
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[order_closed_topology α]
[topological_space β]
{a b : α}
{f : α → β}
(h : a < b) :
continuous_within_at f (set.Ioc a b) b ↔ continuous_within_at f (set.Iic b) b
theorem
frontier_le_subset_eq
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{f g : β → α}
[topological_space β]
(hf : continuous f)
(hg : continuous g) :
theorem
frontier_Iic_subset
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
(a : α) :
theorem
frontier_Ici_subset
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
(a : α) :
theorem
frontier_lt_subset_eq
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{f g : β → α}
[topological_space β]
(hf : continuous f)
(hg : continuous g) :
theorem
continuous_if_le
{α : Type u}
{β : Type v}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{f g : β → α}
[topological_space β]
[topological_space γ]
[Π (x : β), decidable (f x ≤ g x)]
{f' g' : β → γ}
(hf : continuous f)
(hg : continuous g)
(hf' : continuous_on f' {x : β | f x ≤ g x})
(hg' : continuous_on g' {x : β | g x ≤ f x})
(hfg : ∀ (x : β), f x = g x → f' x = g' x) :
continuous (λ (x : β), ite (f x ≤ g x) (f' x) (g' x))
theorem
continuous.if_le
{α : Type u}
{β : Type v}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{f g : β → α}
[topological_space β]
[topological_space γ]
[Π (x : β), decidable (f x ≤ g x)]
{f' g' : β → γ}
(hf' : continuous f')
(hg' : continuous g')
(hf : continuous f)
(hg : continuous g)
(hfg : ∀ (x : β), f x = g x → f' x = g' x) :
continuous (λ (x : β), ite (f x ≤ g x) (f' x) (g' x))
@[continuity]
theorem
continuous.min
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{f g : β → α}
[topological_space β]
(hf : continuous f)
(hg : continuous g) :
continuous (λ (b : β), min (f b) (g b))
@[continuity]
theorem
continuous.max
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{f g : β → α}
[topological_space β]
(hf : continuous f)
(hg : continuous g) :
continuous (λ (b : β), max (f b) (g b))
theorem
continuous_min
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α] :
continuous (λ (p : α × α), min p.fst p.snd)
theorem
continuous_max
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α] :
continuous (λ (p : α × α), max p.fst p.snd)
theorem
filter.tendsto.max
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{f g : β → α}
{b : filter β}
{a₁ a₂ : α}
(hf : filter.tendsto f b (𝓝 a₁))
(hg : filter.tendsto g b (𝓝 a₂)) :
filter.tendsto (λ (b : β), max (f b) (g b)) b (𝓝 (max a₁ a₂))
theorem
filter.tendsto.min
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[order_closed_topology α]
{f g : β → α}
{b : filter β}
{a₁ a₂ : α}
(hf : filter.tendsto f b (𝓝 a₁))
(hg : filter.tendsto g b (𝓝 a₂)) :
filter.tendsto (λ (b : β), min (f b) (g b)) b (𝓝 (min a₁ a₂))
@[class]
- topology_eq_generate_intervals : t = topological_space.generate_from {s : set α | ∃ (a : α), s = set.Ioi a ∨ s = set.Iio a}
The order topology on an ordered type is the topology generated by open intervals. We register
it on a preorder, but it is mostly interesting in linear orders, where it is also order-closed.
We define it as a mixin. If you want to introduce the order topology on a preorder, use
preorder.topology.
(Order) topology on a partial order α generated by the subbase of open intervals
(a, ∞) = { x ∣ a < x }, (-∞ , b) = {x ∣ x < b} for all a, b in α. We do not register it as an
instance as many ordered sets are already endowed with the same topology, most often in a non-defeq
way though. Register as a local instance when necessary.
Equations
- preorder.topology α = topological_space.generate_from {s : set α | ∃ (a : α), s = {b : α | a < b} ∨ s = {b : α | b < a}}
@[protected, instance]
def
order_dual.order_topology
{α : Type u_1}
[topological_space α]
[partial_order α]
[order_topology α] :
theorem
is_open_iff_generate_intervals
{α : Type u}
[topological_space α]
[partial_order α]
[t : order_topology α]
{s : set α} :
theorem
is_open_lt'
{α : Type u}
[topological_space α]
[partial_order α]
[t : order_topology α]
(a : α) :
theorem
is_open_gt'
{α : Type u}
[topological_space α]
[partial_order α]
[t : order_topology α]
(a : α) :
theorem
lt_mem_nhds
{α : Type u}
[topological_space α]
[partial_order α]
[t : order_topology α]
{a b : α}
(h : a < b) :
theorem
le_mem_nhds
{α : Type u}
[topological_space α]
[partial_order α]
[t : order_topology α]
{a b : α}
(h : a < b) :
theorem
gt_mem_nhds
{α : Type u}
[topological_space α]
[partial_order α]
[t : order_topology α]
{a b : α}
(h : a < b) :
theorem
ge_mem_nhds
{α : Type u}
[topological_space α]
[partial_order α]
[t : order_topology α]
{a b : α}
(h : a < b) :
theorem
nhds_eq_order
{α : Type u}
[topological_space α]
[partial_order α]
[t : order_topology α]
(a : α) :
theorem
tendsto_order
{α : Type u}
{β : Type v}
[topological_space α]
[partial_order α]
[t : order_topology α]
{f : β → α}
{a : α}
{x : filter β} :
@[protected, instance]
def
tendsto_Icc_class_nhds
{α : Type u}
[topological_space α]
[partial_order α]
[t : order_topology α]
(a : α) :
filter.tendsto_Ixx_class set.Icc (𝓝 a) (𝓝 a)
@[protected, instance]
def
tendsto_Ico_class_nhds
{α : Type u}
[topological_space α]
[partial_order α]
[t : order_topology α]
(a : α) :
filter.tendsto_Ixx_class set.Ico (𝓝 a) (𝓝 a)
@[protected, instance]
def
tendsto_Ioc_class_nhds
{α : Type u}
[topological_space α]
[partial_order α]
[t : order_topology α]
(a : α) :
filter.tendsto_Ixx_class set.Ioc (𝓝 a) (𝓝 a)
@[protected, instance]
def
tendsto_Ioo_class_nhds
{α : Type u}
[topological_space α]
[partial_order α]
[t : order_topology α]
(a : α) :
filter.tendsto_Ixx_class set.Ioo (𝓝 a) (𝓝 a)
theorem
tendsto_of_tendsto_of_tendsto_of_le_of_le'
{α : Type u}
{β : Type v}
[topological_space α]
[partial_order α]
[t : order_topology α]
{f g h : β → α}
{b : filter β}
{a : α}
(hg : filter.tendsto g b (𝓝 a))
(hh : filter.tendsto h b (𝓝 a))
(hgf : ∀ᶠ (b : β) in b, g b ≤ f b)
(hfh : ∀ᶠ (b : β) in b, f b ≤ h b) :
filter.tendsto f b (𝓝 a)
Also known as squeeze or sandwich theorem. This version assumes that inequalities hold eventually for the filter.
theorem
tendsto_of_tendsto_of_tendsto_of_le_of_le
{α : Type u}
{β : Type v}
[topological_space α]
[partial_order α]
[t : order_topology α]
{f g h : β → α}
{b : filter β}
{a : α}
(hg : filter.tendsto g b (𝓝 a))
(hh : filter.tendsto h b (𝓝 a))
(hgf : g ≤ f)
(hfh : f ≤ h) :
filter.tendsto f b (𝓝 a)
Also known as squeeze or sandwich theorem. This version assumes that inequalities hold everywhere.
theorem
nhds_order_unbounded
{α : Type u}
[topological_space α]
[partial_order α]
[t : order_topology α]
{a : α}
(hu : ∃ (u : α), a < u)
(hl : ∃ (l : α), l < a) :
theorem
tendsto_order_unbounded
{α : Type u}
{β : Type v}
[topological_space α]
[partial_order α]
[t : order_topology α]
{f : β → α}
{a : α}
{x : filter β}
(hu : ∃ (u : α), a < u)
(hl : ∃ (l : α), l < a)
(h : ∀ (l u : α), l < a → a < u → (∀ᶠ (b : β) in x, l < f b ∧ f b < u)) :
filter.tendsto f x (𝓝 a)
@[protected, instance]
def
tendsto_Ixx_nhds_within
{α : Type u_1}
[preorder α]
[topological_space α]
(a : α)
{s t : set α}
{Ixx : α → α → set α}
[filter.tendsto_Ixx_class Ixx (𝓝 a) (𝓝 a)]
[filter.tendsto_Ixx_class Ixx (𝓟 s) (𝓟 t)] :
filter.tendsto_Ixx_class Ixx (𝓝[s] a) (𝓝[t] a)
@[protected, instance]
def
tendsto_Icc_class_nhds_pi
{ι : Type u_1}
{α : ι → Type u_2}
[Π (i : ι), partial_order (α i)]
[Π (i : ι), topological_space (α i)]
[∀ (i : ι), order_topology (α i)]
(f : Π (i : ι), α i) :
filter.tendsto_Ixx_class set.Icc (𝓝 f) (𝓝 f)
theorem
induced_order_topology'
{α : Type u}
{β : Type v}
[partial_order α]
[ta : topological_space β]
[partial_order β]
[order_topology β]
(f : α → β)
(hf : ∀ {x y : α}, f x < f y ↔ x < y)
(H₁ : ∀ {a : α} {x : β}, x < f a → (∃ (b : α) (H : b < a), x ≤ f b))
(H₂ : ∀ {a : α} {x : β}, f a < x → (∃ (b : α) (H : b > a), f b ≤ x)) :
theorem
induced_order_topology
{α : Type u}
{β : Type v}
[partial_order α]
[ta : topological_space β]
[partial_order β]
[order_topology β]
(f : α → β)
(hf : ∀ {x y : α}, f x < f y ↔ x < y)
(H : ∀ {x y : β}, x < y → (∃ (a : α), x < f a ∧ f a < y)) :
@[protected, instance]
def
order_topology_of_ord_connected
{α : Type u}
[ta : topological_space α]
[linear_order α]
[order_topology α]
{t : set α}
[ht : t.ord_connected] :
On an ord_connected subset of a linear order, the order topology for the restriction of the
order is the same as the restriction to the subset of the order topology.
theorem
tendsto_nhds_top_mono
{α : Type u}
{β : Type v}
[topological_space β]
[order_top β]
[order_topology β]
{l : filter α}
{f g : α → β}
(hf : filter.tendsto f l (𝓝 ⊤))
(hg : f ≤ᶠ[l] g) :
filter.tendsto g l (𝓝 ⊤)
theorem
tendsto_nhds_bot_mono
{α : Type u}
{β : Type v}
[topological_space β]
[order_bot β]
[order_topology β]
{l : filter α}
{f g : α → β}
(hf : filter.tendsto f l (𝓝 ⊥))
(hg : g ≤ᶠ[l] f) :
filter.tendsto g l (𝓝 ⊥)
theorem
tendsto_nhds_top_mono'
{α : Type u}
{β : Type v}
[topological_space β]
[order_top β]
[order_topology β]
{l : filter α}
{f g : α → β}
(hf : filter.tendsto f l (𝓝 ⊤))
(hg : f ≤ g) :
filter.tendsto g l (𝓝 ⊤)
theorem
tendsto_nhds_bot_mono'
{α : Type u}
{β : Type v}
[topological_space β]
[order_bot β]
[order_topology β]
{l : filter α}
{f g : α → β}
(hf : filter.tendsto f l (𝓝 ⊥))
(hg : g ≤ f) :
filter.tendsto g l (𝓝 ⊥)
theorem
exists_Ioc_subset_of_mem_nhds'
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a : α}
{s : set α}
(hs : s ∈ 𝓝 a)
{l : α}
(hl : l < a) :
theorem
exists_Ico_subset_of_mem_nhds'
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a : α}
{s : set α}
(hs : s ∈ 𝓝 a)
{u : α}
(hu : a < u) :
theorem
exists_Ioc_subset_of_mem_nhds
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a : α}
{s : set α}
(hs : s ∈ 𝓝 a)
(h : ∃ (l : α), l < a) :
theorem
exists_Ico_subset_of_mem_nhds
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a : α}
{s : set α}
(hs : s ∈ 𝓝 a)
(h : ∃ (u : α), a < u) :
theorem
order_separated
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a₁ a₂ : α}
(h : a₁ < a₂) :
@[protected, instance]
def
order_topology.to_order_closed_topology
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α] :
theorem
order_topology.t2_space
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α] :
t2_space α
@[protected, instance]
def
order_topology.regular_space
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α] :
theorem
mem_nhds_iff_exists_Ioo_subset'
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a : α}
{s : set α}
(hl : ∃ (l : α), l < a)
(hu : ∃ (u : α), a < u) :
A set is a neighborhood of a if and only if it contains an interval (l, u) containing a,
provided a is neither a bottom element nor a top element.
theorem
mem_nhds_iff_exists_Ioo_subset
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[no_top_order α]
[no_bot_order α]
{a : α}
{s : set α} :
A set is a neighborhood of a if and only if it contains an interval (l, u) containing a.
theorem
nhds_basis_Ioo'
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a : α}
(hl : ∃ (l : α), l < a)
(hu : ∃ (u : α), a < u) :
theorem
nhds_basis_Ioo
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[no_top_order α]
[no_bot_order α]
{a : α} :
theorem
filter.eventually.exists_Ioo_subset
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[no_top_order α]
[no_bot_order α]
{a : α}
{p : α → Prop}
(hp : ∀ᶠ (x : α) in 𝓝 a, p x) :
theorem
Iio_mem_nhds
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a b : α}
(h : a < b) :
theorem
Ioi_mem_nhds
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a b : α}
(h : a < b) :
theorem
Iic_mem_nhds
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a b : α}
(h : a < b) :
theorem
Ici_mem_nhds
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a b : α}
(h : a < b) :
theorem
Ioo_mem_nhds
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a b x : α}
(ha : a < x)
(hb : x < b) :
theorem
Ioc_mem_nhds
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a b x : α}
(ha : a < x)
(hb : x < b) :
theorem
Ico_mem_nhds
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a b x : α}
(ha : a < x)
(hb : x < b) :
theorem
Icc_mem_nhds
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a b x : α}
(ha : a < x)
(hb : x < b) :
Intervals in Π i, π i belong to 𝓝 x #
For each leamma pi_Ixx_mem_nhds we add a non-dependent version pi_Ixx_mem_nhds' because
sometimes Lean fails to unify different instances while trying to apply the dependent version to,
e.g., ι → ℝ.
theorem
pi_Iic_mem_nhds
{ι : Type u_1}
{π : ι → Type u_2}
[fintype ι]
[Π (i : ι), linear_order («π» i)]
[Π (i : ι), topological_space («π» i)]
[∀ (i : ι), order_topology («π» i)]
{a x : Π (i : ι), «π» i}
(ha : ∀ (i : ι), x i < a i) :
theorem
pi_Iic_mem_nhds'
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{ι : Type u_1}
[fintype ι]
{a' x' : ι → α}
(ha : ∀ (i : ι), x' i < a' i) :
theorem
pi_Ici_mem_nhds
{ι : Type u_1}
{π : ι → Type u_2}
[fintype ι]
[Π (i : ι), linear_order («π» i)]
[Π (i : ι), topological_space («π» i)]
[∀ (i : ι), order_topology («π» i)]
{a x : Π (i : ι), «π» i}
(ha : ∀ (i : ι), a i < x i) :
theorem
pi_Ici_mem_nhds'
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{ι : Type u_1}
[fintype ι]
{a' x' : ι → α}
(ha : ∀ (i : ι), a' i < x' i) :
theorem
pi_Icc_mem_nhds
{ι : Type u_1}
{π : ι → Type u_2}
[fintype ι]
[Π (i : ι), linear_order («π» i)]
[Π (i : ι), topological_space («π» i)]
[∀ (i : ι), order_topology («π» i)]
{a b x : Π (i : ι), «π» i}
(ha : ∀ (i : ι), a i < x i)
(hb : ∀ (i : ι), x i < b i) :
theorem
pi_Icc_mem_nhds'
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{ι : Type u_1}
[fintype ι]
{a' b' x' : ι → α}
(ha : ∀ (i : ι), a' i < x' i)
(hb : ∀ (i : ι), x' i < b' i) :
theorem
pi_Iio_mem_nhds
{ι : Type u_1}
{π : ι → Type u_2}
[fintype ι]
[Π (i : ι), linear_order («π» i)]
[Π (i : ι), topological_space («π» i)]
[∀ (i : ι), order_topology («π» i)]
{a x : Π (i : ι), «π» i}
[nonempty ι]
(ha : ∀ (i : ι), x i < a i) :
theorem
pi_Iio_mem_nhds'
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{ι : Type u_1}
[fintype ι]
{a' x' : ι → α}
[nonempty ι]
(ha : ∀ (i : ι), x' i < a' i) :
theorem
pi_Ioi_mem_nhds
{ι : Type u_1}
{π : ι → Type u_2}
[fintype ι]
[Π (i : ι), linear_order («π» i)]
[Π (i : ι), topological_space («π» i)]
[∀ (i : ι), order_topology («π» i)]
{a x : Π (i : ι), «π» i}
[nonempty ι]
(ha : ∀ (i : ι), a i < x i) :
theorem
pi_Ioi_mem_nhds'
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{ι : Type u_1}
[fintype ι]
{a' x' : ι → α}
[nonempty ι]
(ha : ∀ (i : ι), a' i < x' i) :
theorem
pi_Ioc_mem_nhds
{ι : Type u_1}
{π : ι → Type u_2}
[fintype ι]
[Π (i : ι), linear_order («π» i)]
[Π (i : ι), topological_space («π» i)]
[∀ (i : ι), order_topology («π» i)]
{a b x : Π (i : ι), «π» i}
[nonempty ι]
(ha : ∀ (i : ι), a i < x i)
(hb : ∀ (i : ι), x i < b i) :
theorem
pi_Ioc_mem_nhds'
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{ι : Type u_1}
[fintype ι]
{a' b' x' : ι → α}
[nonempty ι]
(ha : ∀ (i : ι), a' i < x' i)
(hb : ∀ (i : ι), x' i < b' i) :
theorem
pi_Ico_mem_nhds
{ι : Type u_1}
{π : ι → Type u_2}
[fintype ι]
[Π (i : ι), linear_order («π» i)]
[Π (i : ι), topological_space («π» i)]
[∀ (i : ι), order_topology («π» i)]
{a b x : Π (i : ι), «π» i}
[nonempty ι]
(ha : ∀ (i : ι), a i < x i)
(hb : ∀ (i : ι), x i < b i) :
theorem
pi_Ico_mem_nhds'
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{ι : Type u_1}
[fintype ι]
{a' b' x' : ι → α}
[nonempty ι]
(ha : ∀ (i : ι), a' i < x' i)
(hb : ∀ (i : ι), x' i < b' i) :
theorem
pi_Ioo_mem_nhds
{ι : Type u_1}
{π : ι → Type u_2}
[fintype ι]
[Π (i : ι), linear_order («π» i)]
[Π (i : ι), topological_space («π» i)]
[∀ (i : ι), order_topology («π» i)]
{a b x : Π (i : ι), «π» i}
[nonempty ι]
(ha : ∀ (i : ι), a i < x i)
(hb : ∀ (i : ι), x i < b i) :
theorem
pi_Ioo_mem_nhds'
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{ι : Type u_1}
[fintype ι]
{a' b' x' : ι → α}
[nonempty ι]
(ha : ∀ (i : ι), a' i < x' i)
(hb : ∀ (i : ι), x' i < b' i) :
theorem
disjoint_nhds_at_top
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[no_top_order α]
(x : α) :
disjoint (𝓝 x) filter.at_top
@[simp]
theorem
inf_nhds_at_top
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[no_top_order α]
(x : α) :
𝓝 x ⊓ filter.at_top = ⊥
theorem
disjoint_nhds_at_bot
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[no_bot_order α]
(x : α) :
disjoint (𝓝 x) filter.at_bot
@[simp]
theorem
inf_nhds_at_bot
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[no_bot_order α]
(x : α) :
𝓝 x ⊓ filter.at_bot = ⊥
theorem
not_tendsto_nhds_of_tendsto_at_top
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[order_topology α]
[no_top_order α]
{F : filter β}
[F.ne_bot]
{f : β → α}
(hf : filter.tendsto f F filter.at_top)
(x : α) :
¬filter.tendsto f F (𝓝 x)
theorem
not_tendsto_at_top_of_tendsto_nhds
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[order_topology α]
[no_top_order α]
{F : filter β}
[F.ne_bot]
{f : β → α}
{x : α}
(hf : filter.tendsto f F (𝓝 x)) :
theorem
not_tendsto_nhds_of_tendsto_at_bot
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[order_topology α]
[no_bot_order α]
{F : filter β}
[F.ne_bot]
{f : β → α}
(hf : filter.tendsto f F filter.at_bot)
(x : α) :
¬filter.tendsto f F (𝓝 x)
theorem
not_tendsto_at_bot_of_tendsto_nhds
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[order_topology α]
[no_bot_order α]
{F : filter β}
[F.ne_bot]
{f : β → α}
{x : α}
(hf : filter.tendsto f F (𝓝 x)) :
Neighborhoods to the left and to the right on an order_topology #
We've seen some properties of left and right neighborhood of a point in an order_closed_topology.
In an order_topology, such neighborhoods can be characterized as the sets containing suitable
intervals to the right or to the left of a. We give now these characterizations.
theorem
tfae_mem_nhds_within_Ioi
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a b : α}
(hab : a < b)
(s : set α) :
The following statements are equivalent:
sis a neighborhood ofawithin(a, +∞)sis a neighborhood ofawithin(a, b]sis a neighborhood ofawithin(a, b)sincludes(a, u)for someu ∈ (a, b]sincludes(a, u)for someu > a
theorem
mem_nhds_within_Ioi_iff_exists_mem_Ioc_Ioo_subset
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a u' : α}
{s : set α}
(hu' : a < u') :
theorem
mem_nhds_within_Ioi_iff_exists_Ioo_subset'
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a u' : α}
{s : set α}
(hu' : a < u') :
A set is a neighborhood of a within (a, +∞) if and only if it contains an interval (a, u)
with a < u < u', provided a is not a top element.
theorem
mem_nhds_within_Ioi_iff_exists_Ioo_subset
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[no_top_order α]
{a : α}
{s : set α} :
A set is a neighborhood of a within (a, +∞) if and only if it contains an interval (a, u)
with a < u.
theorem
mem_nhds_within_Ioi_iff_exists_Ioc_subset
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[no_top_order α]
[densely_ordered α]
{a : α}
{s : set α} :
A set is a neighborhood of a within (a, +∞) if and only if it contains an interval (a, u]
with a < u.
theorem
tfae_mem_nhds_within_Iio
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a b : α}
(h : a < b)
(s : set α) :
The following statements are equivalent:
sis a neighborhood ofbwithin(-∞, b)sis a neighborhood ofbwithin[a, b)sis a neighborhood ofbwithin(a, b)sincludes(l, b)for somel ∈ [a, b)sincludes(l, b)for somel < b
theorem
mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a l' : α}
{s : set α}
(hl' : l' < a) :
theorem
mem_nhds_within_Iio_iff_exists_Ioo_subset'
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a l' : α}
{s : set α}
(hl' : l' < a) :
A set is a neighborhood of a within (-∞, a) if and only if it contains an interval (l, a)
with l < a, provided a is not a bottom element.
theorem
mem_nhds_within_Iio_iff_exists_Ioo_subset
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[no_bot_order α]
{a : α}
{s : set α} :
A set is a neighborhood of a within (-∞, a) if and only if it contains an interval (l, a)
with l < a.
theorem
mem_nhds_within_Iio_iff_exists_Ico_subset
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[no_bot_order α]
[densely_ordered α]
{a : α}
{s : set α} :
A set is a neighborhood of a within (-∞, a) if and only if it contains an interval [l, a)
with l < a.
theorem
tfae_mem_nhds_within_Ici
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a b : α}
(hab : a < b)
(s : set α) :
The following statements are equivalent:
sis a neighborhood ofawithin[a, +∞)sis a neighborhood ofawithin[a, b]sis a neighborhood ofawithin[a, b)sincludes[a, u)for someu ∈ (a, b]sincludes[a, u)for someu > a
theorem
mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a u' : α}
{s : set α}
(hu' : a < u') :
theorem
mem_nhds_within_Ici_iff_exists_Ico_subset'
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a u' : α}
{s : set α}
(hu' : a < u') :
A set is a neighborhood of a within [a, +∞) if and only if it contains an interval [a, u)
with a < u < u', provided a is not a top element.
theorem
mem_nhds_within_Ici_iff_exists_Ico_subset
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[no_top_order α]
{a : α}
{s : set α} :
A set is a neighborhood of a within [a, +∞) if and only if it contains an interval [a, u)
with a < u.
theorem
mem_nhds_within_Ici_iff_exists_Icc_subset'
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[no_top_order α]
[densely_ordered α]
{a : α}
{s : set α} :
A set is a neighborhood of a within [a, +∞) if and only if it contains an interval [a, u]
with a < u.
theorem
tfae_mem_nhds_within_Iic
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a b : α}
(h : a < b)
(s : set α) :
The following statements are equivalent:
sis a neighborhood ofbwithin(-∞, b]sis a neighborhood ofbwithin[a, b]sis a neighborhood ofbwithin(a, b]sincludes(l, b]for somel ∈ [a, b)sincludes(l, b]for somel < b
theorem
mem_nhds_within_Iic_iff_exists_mem_Ico_Ioc_subset
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a l' : α}
{s : set α}
(hl' : l' < a) :
theorem
mem_nhds_within_Iic_iff_exists_Ioc_subset'
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a l' : α}
{s : set α}
(hl' : l' < a) :
A set is a neighborhood of a within (-∞, a] if and only if it contains an interval (l, a]
with l < a, provided a is not a bottom element.
theorem
mem_nhds_within_Iic_iff_exists_Ioc_subset
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[no_bot_order α]
{a : α}
{s : set α} :
A set is a neighborhood of a within (-∞, a] if and only if it contains an interval (l, a]
with l < a.
theorem
mem_nhds_within_Iic_iff_exists_Icc_subset'
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[no_bot_order α]
[densely_ordered α]
{a : α}
{s : set α} :
A set is a neighborhood of a within (-∞, a] if and only if it contains an interval [l, a]
with l < a.
theorem
mem_nhds_within_Ici_iff_exists_Icc_subset
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[no_top_order α]
[densely_ordered α]
{a : α}
{s : set α} :
A set is a neighborhood of a within [a, +∞) if and only if it contains an interval [a, u]
with a < u.
theorem
mem_nhds_within_Iic_iff_exists_Icc_subset
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[no_bot_order α]
[densely_ordered α]
{a : α}
{s : set α} :
A set is a neighborhood of a within (-∞, a] if and only if it contains an interval [l, a]
with l < a.
theorem
nhds_eq_infi_abs_sub
{α : Type u}
[topological_space α]
[linear_ordered_add_comm_group α]
[order_topology α]
(a : α) :
theorem
order_topology_of_nhds_abs
{α : Type u_1}
[topological_space α]
[linear_ordered_add_comm_group α]
(h_nhds : ∀ (a : α), 𝓝 a = ⨅ (r : α) (H : r > 0), 𝓟 {b : α | abs (a - b) < r}) :
theorem
linear_ordered_add_comm_group.tendsto_nhds
{α : Type u}
{β : Type v}
[topological_space α]
[linear_ordered_add_comm_group α]
[order_topology α]
{f : β → α}
{x : filter β}
{a : α} :
theorem
eventually_abs_sub_lt
{α : Type u}
[topological_space α]
[linear_ordered_add_comm_group α]
[order_topology α]
(a : α)
{ε : α}
(hε : 0 < ε) :
@[protected, instance]
def
linear_ordered_add_comm_group.topological_add_group
{α : Type u}
[topological_space α]
[linear_ordered_add_comm_group α]
[order_topology α] :
@[continuity]
theorem
continuous_abs
{α : Type u}
[topological_space α]
[linear_ordered_add_comm_group α]
[order_topology α] :
theorem
filter.tendsto.abs
{α : Type u}
{β : Type v}
[topological_space α]
[linear_ordered_add_comm_group α]
[order_topology α]
{f : β → α}
{a : α}
{l : filter β}
(h : filter.tendsto f l (𝓝 a)) :
filter.tendsto (λ (x : β), abs (f x)) l (𝓝 (abs a))
theorem
continuous.abs
{α : Type u}
{β : Type v}
[topological_space α]
[linear_ordered_add_comm_group α]
[order_topology α]
{f : β → α}
[topological_space β]
(h : continuous f) :
continuous (λ (x : β), abs (f x))
theorem
continuous_at.abs
{α : Type u}
{β : Type v}
[topological_space α]
[linear_ordered_add_comm_group α]
[order_topology α]
{f : β → α}
[topological_space β]
{b : β}
(h : continuous_at f b) :
continuous_at (λ (x : β), abs (f x)) b
theorem
continuous_within_at.abs
{α : Type u}
{β : Type v}
[topological_space α]
[linear_ordered_add_comm_group α]
[order_topology α]
{f : β → α}
[topological_space β]
{b : β}
{s : set β}
(h : continuous_within_at f s b) :
continuous_within_at (λ (x : β), abs (f x)) s b
theorem
continuous_on.abs
{α : Type u}
{β : Type v}
[topological_space α]
[linear_ordered_add_comm_group α]
[order_topology α]
{f : β → α}
[topological_space β]
{s : set β}
(h : continuous_on f s) :
continuous_on (λ (x : β), abs (f x)) s
theorem
tendsto_abs_nhds_within_zero
{α : Type u}
[topological_space α]
[linear_ordered_add_comm_group α]
[order_topology α] :
theorem
filter.tendsto.add_at_top
{α : Type u}
{β : Type v}
[topological_space α]
[linear_ordered_add_comm_group α]
[order_topology α]
{l : filter β}
{f g : β → α}
{C : α}
(hf : filter.tendsto f l (𝓝 C))
(hg : filter.tendsto g l filter.at_top) :
filter.tendsto (λ (x : β), f x + g x) l filter.at_top
In a linearly ordered additive commutative group with the order topology, if f tends to C
and g tends to at_top then f + g tends to at_top.
theorem
filter.tendsto.add_at_bot
{α : Type u}
{β : Type v}
[topological_space α]
[linear_ordered_add_comm_group α]
[order_topology α]
{l : filter β}
{f g : β → α}
{C : α}
(hf : filter.tendsto f l (𝓝 C))
(hg : filter.tendsto g l filter.at_bot) :
filter.tendsto (λ (x : β), f x + g x) l filter.at_bot
In a linearly ordered additive commutative group with the order topology, if f tends to C
and g tends to at_bot then f + g tends to at_bot.
theorem
filter.tendsto.at_top_add
{α : Type u}
{β : Type v}
[topological_space α]
[linear_ordered_add_comm_group α]
[order_topology α]
{l : filter β}
{f g : β → α}
{C : α}
(hf : filter.tendsto f l filter.at_top)
(hg : filter.tendsto g l (𝓝 C)) :
filter.tendsto (λ (x : β), f x + g x) l filter.at_top
In a linearly ordered additive commutative group with the order topology, if f tends to
at_top and g tends to C then f + g tends to at_top.
theorem
filter.tendsto.at_bot_add
{α : Type u}
{β : Type v}
[topological_space α]
[linear_ordered_add_comm_group α]
[order_topology α]
{l : filter β}
{f g : β → α}
{C : α}
(hf : filter.tendsto f l filter.at_bot)
(hg : filter.tendsto g l (𝓝 C)) :
filter.tendsto (λ (x : β), f x + g x) l filter.at_bot
In a linearly ordered additive commutative group with the order topology, if f tends to
at_bot and g tends to C then f + g tends to at_bot.
theorem
filter.tendsto.at_top_mul
{α : Type u}
{β : Type v}
[linear_ordered_field α]
[topological_space α]
[order_topology α]
{l : filter β}
{f g : β → α}
{C : α}
(hC : 0 < C)
(hf : filter.tendsto f l filter.at_top)
(hg : filter.tendsto g l (𝓝 C)) :
filter.tendsto (λ (x : β), (f x) * g x) l filter.at_top
In a linearly ordered field with the order topology, if f tends to at_top and g tends to
a positive constant C then f * g tends to at_top.
theorem
filter.tendsto.mul_at_top
{α : Type u}
{β : Type v}
[linear_ordered_field α]
[topological_space α]
[order_topology α]
{l : filter β}
{f g : β → α}
{C : α}
(hC : 0 < C)
(hf : filter.tendsto f l (𝓝 C))
(hg : filter.tendsto g l filter.at_top) :
filter.tendsto (λ (x : β), (f x) * g x) l filter.at_top
In a linearly ordered field with the order topology, if f tends to a positive constant C and
g tends to at_top then f * g tends to at_top.
theorem
filter.tendsto.at_top_mul_neg
{α : Type u}
{β : Type v}
[linear_ordered_field α]
[topological_space α]
[order_topology α]
{l : filter β}
{f g : β → α}
{C : α}
(hC : C < 0)
(hf : filter.tendsto f l filter.at_top)
(hg : filter.tendsto g l (𝓝 C)) :
filter.tendsto (λ (x : β), (f x) * g x) l filter.at_bot
In a linearly ordered field with the order topology, if f tends to at_top and g tends to
a negative constant C then f * g tends to at_bot.
theorem
filter.tendsto.neg_mul_at_top
{α : Type u}
{β : Type v}
[linear_ordered_field α]
[topological_space α]
[order_topology α]
{l : filter β}
{f g : β → α}
{C : α}
(hC : C < 0)
(hf : filter.tendsto f l (𝓝 C))
(hg : filter.tendsto g l filter.at_top) :
filter.tendsto (λ (x : β), (f x) * g x) l filter.at_bot
In a linearly ordered field with the order topology, if f tends to a negative constant C and
g tends to at_top then f * g tends to at_bot.
theorem
filter.tendsto.at_bot_mul
{α : Type u}
{β : Type v}
[linear_ordered_field α]
[topological_space α]
[order_topology α]
{l : filter β}
{f g : β → α}
{C : α}
(hC : 0 < C)
(hf : filter.tendsto f l filter.at_bot)
(hg : filter.tendsto g l (𝓝 C)) :
filter.tendsto (λ (x : β), (f x) * g x) l filter.at_bot
In a linearly ordered field with the order topology, if f tends to at_bot and g tends to
a positive constant C then f * g tends to at_bot.
theorem
filter.tendsto.at_bot_mul_neg
{α : Type u}
{β : Type v}
[linear_ordered_field α]
[topological_space α]
[order_topology α]
{l : filter β}
{f g : β → α}
{C : α}
(hC : C < 0)
(hf : filter.tendsto f l filter.at_bot)
(hg : filter.tendsto g l (𝓝 C)) :
filter.tendsto (λ (x : β), (f x) * g x) l filter.at_top
In a linearly ordered field with the order topology, if f tends to at_bot and g tends to
a negative constant C then f * g tends to at_top.
theorem
filter.tendsto.mul_at_bot
{α : Type u}
{β : Type v}
[linear_ordered_field α]
[topological_space α]
[order_topology α]
{l : filter β}
{f g : β → α}
{C : α}
(hC : 0 < C)
(hf : filter.tendsto f l (𝓝 C))
(hg : filter.tendsto g l filter.at_bot) :
filter.tendsto (λ (x : β), (f x) * g x) l filter.at_bot
In a linearly ordered field with the order topology, if f tends to a positive constant C and
g tends to at_bot then f * g tends to at_bot.
theorem
filter.tendsto.neg_mul_at_bot
{α : Type u}
{β : Type v}
[linear_ordered_field α]
[topological_space α]
[order_topology α]
{l : filter β}
{f g : β → α}
{C : α}
(hC : C < 0)
(hf : filter.tendsto f l (𝓝 C))
(hg : filter.tendsto g l filter.at_bot) :
filter.tendsto (λ (x : β), (f x) * g x) l filter.at_top
In a linearly ordered field with the order topology, if f tends to a negative constant C and
g tends to at_bot then f * g tends to at_top.
theorem
tendsto_inv_zero_at_top
{α : Type u}
[linear_ordered_field α]
[topological_space α]
[order_topology α] :
filter.tendsto (λ (x : α), x⁻¹) (𝓝[set.Ioi 0] 0) filter.at_top
The function x ↦ x⁻¹ tends to +∞ on the right of 0.
theorem
tendsto_inv_at_top_zero'
{α : Type u}
[linear_ordered_field α]
[topological_space α]
[order_topology α] :
filter.tendsto (λ (r : α), r⁻¹) filter.at_top (𝓝[set.Ioi 0] 0)
The function r ↦ r⁻¹ tends to 0 on the right as r → +∞.
theorem
tendsto_inv_at_top_zero
{α : Type u}
[linear_ordered_field α]
[topological_space α]
[order_topology α] :
filter.tendsto (λ (r : α), r⁻¹) filter.at_top (𝓝 0)
theorem
filter.tendsto.div_at_top
{α : Type u}
{β : Type v}
[linear_ordered_field α]
[topological_space α]
[order_topology α]
[has_continuous_mul α]
{f g : β → α}
{l : filter β}
{a : α}
(h : filter.tendsto f l (𝓝 a))
(hg : filter.tendsto g l filter.at_top) :
filter.tendsto (λ (x : β), f x / g x) l (𝓝 0)
theorem
filter.tendsto.inv_tendsto_at_top
{α : Type u}
{β : Type v}
[linear_ordered_field α]
[topological_space α]
[order_topology α]
{l : filter β}
{f : β → α}
(h : filter.tendsto f l filter.at_top) :
filter.tendsto f⁻¹ l (𝓝 0)
theorem
filter.tendsto.inv_tendsto_zero
{α : Type u}
{β : Type v}
[linear_ordered_field α]
[topological_space α]
[order_topology α]
{l : filter β}
{f : β → α}
(h : filter.tendsto f l (𝓝[set.Ioi 0] 0)) :
theorem
tendsto_pow_neg_at_top
{α : Type u}
[linear_ordered_field α]
[topological_space α]
[order_topology α]
{n : ℕ}
(hn : 1 ≤ n) :
filter.tendsto (λ (x : α), x ^ -↑n) filter.at_top (𝓝 0)
The function x^(-n) tends to 0 at +∞ for any positive natural n.
A version for positive real powers exists as tendsto_rpow_neg_at_top.
theorem
tendsto_fpow_at_top_zero
{α : Type u}
[linear_ordered_field α]
[topological_space α]
[order_topology α]
{n : ℤ}
(hn : n < 0) :
filter.tendsto (λ (x : α), x ^ n) filter.at_top (𝓝 0)
theorem
is_lub.frequently_mem
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a : α}
{s : set α}
(ha : is_lub s a)
(hs : s.nonempty) :
theorem
is_lub.frequently_nhds_mem
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a : α}
{s : set α}
(ha : is_lub s a)
(hs : s.nonempty) :
theorem
is_glb.frequently_mem
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a : α}
{s : set α}
(ha : is_glb s a)
(hs : s.nonempty) :
theorem
is_glb.frequently_nhds_mem
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a : α}
{s : set α}
(ha : is_glb s a)
(hs : s.nonempty) :
theorem
is_lub.mem_closure
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a : α}
{s : set α}
(ha : is_lub s a)
(hs : s.nonempty) :
theorem
is_glb.mem_closure
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a : α}
{s : set α}
(ha : is_glb s a)
(hs : s.nonempty) :
theorem
is_lub.nhds_within_ne_bot
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a : α}
{s : set α}
(ha : is_lub s a)
(hs : s.nonempty) :
theorem
is_glb.nhds_within_ne_bot
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a : α}
{s : set α} :
theorem
is_lub_of_mem_nhds
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{s : set α}
{a : α}
{f : filter α}
(hsa : a ∈ upper_bounds s)
(hsf : s ∈ f)
[(f ⊓ 𝓝 a).ne_bot] :
is_lub s a
theorem
is_glb_of_mem_nhds
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{s : set α}
{a : α}
{f : filter α} :
theorem
is_lub.mem_upper_bounds_of_tendsto
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_topology α]
[preorder γ]
[topological_space γ]
[order_closed_topology γ]
{f : α → γ}
{s : set α}
{a : α}
{b : γ}
(hf : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → x ≤ y → f x ≤ f y)
(ha : is_lub s a)
(hb : filter.tendsto f (𝓝[s] a) (𝓝 b)) :
b ∈ upper_bounds (f '' s)
theorem
is_lub.is_lub_of_tendsto
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_topology α]
[preorder γ]
[topological_space γ]
[order_closed_topology γ]
{f : α → γ}
{s : set α}
{a : α}
{b : γ}
(hf : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → x ≤ y → f x ≤ f y)
(ha : is_lub s a)
(hs : s.nonempty)
(hb : filter.tendsto f (𝓝[s] a) (𝓝 b)) :
theorem
is_glb.mem_lower_bounds_of_tendsto
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_topology α]
[preorder γ]
[topological_space γ]
[order_closed_topology γ]
{f : α → γ}
{s : set α}
{a : α}
{b : γ}
(hf : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → x ≤ y → f x ≤ f y)
(ha : is_glb s a)
(hb : filter.tendsto f (𝓝[s] a) (𝓝 b)) :
b ∈ lower_bounds (f '' s)
theorem
is_glb.is_glb_of_tendsto
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_topology α]
[preorder γ]
[topological_space γ]
[order_closed_topology γ]
{f : α → γ}
{s : set α}
{a : α}
{b : γ}
(hf : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → x ≤ y → f x ≤ f y) :
theorem
is_lub.mem_lower_bounds_of_tendsto
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_topology α]
[preorder γ]
[topological_space γ]
[order_closed_topology γ]
{f : α → γ}
{s : set α}
{a : α}
{b : γ}
(hf : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → x ≤ y → f y ≤ f x)
(ha : is_lub s a)
(hb : filter.tendsto f (𝓝[s] a) (𝓝 b)) :
b ∈ lower_bounds (f '' s)
theorem
is_lub.is_glb_of_tendsto
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_topology α]
[preorder γ]
[topological_space γ]
[order_closed_topology γ]
{f : α → γ}
{s : set α}
{a : α}
{b : γ} :
theorem
is_glb.mem_upper_bounds_of_tendsto
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_topology α]
[preorder γ]
[topological_space γ]
[order_closed_topology γ]
{f : α → γ}
{s : set α}
{a : α}
{b : γ}
(hf : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → x ≤ y → f y ≤ f x)
(ha : is_glb s a)
(hb : filter.tendsto f (𝓝[s] a) (𝓝 b)) :
b ∈ upper_bounds (f '' s)
theorem
is_glb.is_lub_of_tendsto
{α : Type u}
{γ : Type w}
[topological_space α]
[linear_order α]
[order_topology α]
[preorder γ]
[topological_space γ]
[order_closed_topology γ]
{f : α → γ}
{s : set α}
{a : α}
{b : γ} :
theorem
is_lub.mem_of_is_closed
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a : α}
{s : set α}
(ha : is_lub s a)
(hs : s.nonempty)
(sc : is_closed s) :
a ∈ s
theorem
is_closed.is_lub_mem
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a : α}
{s : set α}
(ha : is_lub s a)
(hs : s.nonempty)
(sc : is_closed s) :
a ∈ s
Alias of is_lub.mem_of_is_closed.
theorem
is_glb.mem_of_is_closed
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a : α}
{s : set α}
(ha : is_glb s a)
(hs : s.nonempty)
(sc : is_closed s) :
a ∈ s
theorem
is_closed.is_glb_mem
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
{a : α}
{s : set α}
(ha : is_glb s a)
(hs : s.nonempty)
(sc : is_closed s) :
a ∈ s
Alias of is_glb.mem_of_is_closed.
theorem
is_compact.bdd_below
{α : Type u}
[topological_space α]
[linear_order α]
[order_closed_topology α]
[nonempty α]
{s : set α}
(hs : is_compact s) :
A compact set is bounded below
theorem
is_compact.bdd_above
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[nonempty α]
{s : set α} :
is_compact s → bdd_above s
A compact set is bounded above
theorem
closure_Ioi'
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
{a b : α}
(hab : a < b) :
The closure of the interval (a, +∞) is the closed interval [a, +∞), unless a is a top
element.
@[simp]
theorem
closure_Ioi
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
(a : α)
[no_top_order α] :
The closure of the interval (a, +∞) is the closed interval [a, +∞).
theorem
closure_Iio'
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
{a b : α}
(hab : b < a) :
The closure of the interval (-∞, a) is the closed interval (-∞, a], unless a is a bottom
element.
@[simp]
theorem
closure_Iio
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
(a : α)
[no_bot_order α] :
The closure of the interval (-∞, a) is the interval (-∞, a].
@[simp]
theorem
closure_Ioo
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
{a b : α}
(hab : a < b) :
The closure of the open interval (a, b) is the closed interval [a, b].
@[simp]
theorem
closure_Ioc
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
{a b : α}
(hab : a < b) :
The closure of the interval (a, b] is the closed interval [a, b].
@[simp]
theorem
closure_Ico
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
{a b : α}
(hab : a < b) :
The closure of the interval [a, b) is the closed interval [a, b].
@[simp]
theorem
interior_Ici
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
[no_bot_order α]
{a : α} :
@[simp]
theorem
interior_Iic
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
[no_top_order α]
{a : α} :
@[simp]
theorem
interior_Icc
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
[no_bot_order α]
[no_top_order α]
{a b : α} :
@[simp]
theorem
interior_Ico
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
[no_bot_order α]
{a b : α} :
@[simp]
theorem
interior_Ioc
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
[no_top_order α]
{a b : α} :
@[simp]
theorem
frontier_Ici
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
[no_bot_order α]
{a : α} :
@[simp]
theorem
frontier_Iic
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
[no_top_order α]
{a : α} :
@[simp]
theorem
frontier_Ioi
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
[no_top_order α]
{a : α} :
@[simp]
theorem
frontier_Iio
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
[no_bot_order α]
{a : α} :
@[simp]
theorem
frontier_Icc
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
[no_bot_order α]
[no_top_order α]
{a b : α}
(h : a < b) :
@[simp]
theorem
frontier_Ioo
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
{a b : α}
(h : a < b) :
@[simp]
theorem
frontier_Ico
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
[no_bot_order α]
{a b : α}
(h : a < b) :
@[simp]
theorem
frontier_Ioc
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
[no_top_order α]
{a b : α}
(h : a < b) :
theorem
nhds_within_Ioi_ne_bot'
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
{a b c : α}
(H₁ : a < c)
(H₂ : a ≤ b) :
theorem
nhds_within_Ioi_ne_bot
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
[no_top_order α]
{a b : α}
(H : a ≤ b) :
theorem
nhds_within_Ioi_self_ne_bot'
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
{a b : α}
(H : a < b) :
@[instance]
theorem
nhds_within_Ioi_self_ne_bot
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
[no_top_order α]
(a : α) :
theorem
nhds_within_Iio_ne_bot'
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
{a b c : α}
(H₁ : a < c)
(H₂ : b ≤ c) :
theorem
nhds_within_Iio_ne_bot
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
[no_bot_order α]
{a b : α}
(H : a ≤ b) :
theorem
nhds_within_Iio_self_ne_bot'
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
{a b : α}
(H : a < b) :
@[instance]
theorem
nhds_within_Iio_self_ne_bot
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
[no_bot_order α]
(a : α) :
theorem
right_nhds_within_Ico_ne_bot
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
{a b : α}
(H : a < b) :
theorem
left_nhds_within_Ioc_ne_bot
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
{a b : α}
(H : a < b) :
theorem
left_nhds_within_Ioo_ne_bot
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
{a b : α}
(H : a < b) :
theorem
right_nhds_within_Ioo_ne_bot
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
{a b : α}
(H : a < b) :
theorem
comap_coe_nhds_within_Iio_of_Ioo_subset
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
{b : α}
{s : set α}
(hb : s ⊆ set.Iio b)
(hs : s.nonempty → (∃ (a : α) (H : a < b), set.Ioo a b ⊆ s)) :
filter.comap coe (𝓝[set.Iio b] b) = filter.at_top
theorem
comap_coe_nhds_within_Ioi_of_Ioo_subset
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
{a : α}
{s : set α}
(ha : s ⊆ set.Ioi a)
(hs : s.nonempty → (∃ (b : α) (H : b > a), set.Ioo a b ⊆ s)) :
filter.comap coe (𝓝[set.Ioi a] a) = filter.at_bot
theorem
map_coe_at_top_of_Ioo_subset
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
{b : α}
{s : set α}
(hb : s ⊆ set.Iio b)
(hs : ∀ (a' : α), a' < b → (∃ (a : α) (H : a < b), set.Ioo a b ⊆ s)) :
theorem
map_coe_at_bot_of_Ioo_subset
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
{a : α}
{s : set α}
(ha : s ⊆ set.Ioi a)
(hs : ∀ (b' : α), b' > a → (∃ (b : α) (H : b > a), set.Ioo a b ⊆ s)) :
theorem
comap_coe_Ioo_nhds_within_Iio
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
(a b : α) :
filter.comap coe (𝓝[set.Iio b] b) = filter.at_top
The at_top filter for an open interval Ioo a b comes from the left-neighbourhoods filter at
the right endpoint in the ambient order.
theorem
comap_coe_Ioo_nhds_within_Ioi
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
(a b : α) :
filter.comap coe (𝓝[set.Ioi a] a) = filter.at_bot
The at_bot filter for an open interval Ioo a b comes from the right-neighbourhoods filter at
the left endpoint in the ambient order.
theorem
comap_coe_Ioi_nhds_within_Ioi
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
(a : α) :
filter.comap coe (𝓝[set.Ioi a] a) = filter.at_bot
theorem
comap_coe_Iio_nhds_within_Iio
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
(a : α) :
filter.comap coe (𝓝[set.Iio a] a) = filter.at_top
@[simp]
theorem
map_coe_Ioo_at_top
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
{a b : α}
(h : a < b) :
@[simp]
theorem
map_coe_Ioo_at_bot
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
{a b : α}
(h : a < b) :
@[simp]
theorem
map_coe_Ioi_at_bot
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
(a : α) :
@[simp]
theorem
map_coe_Iio_at_top
{α : Type u}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
(a : α) :
@[simp]
theorem
tendsto_comp_coe_Ioo_at_top
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
{a b : α}
{l : filter β}
{f : α → β}
(h : a < b) :
filter.tendsto (λ (x : ↥(set.Ioo a b)), f ↑x) filter.at_top l ↔ filter.tendsto f (𝓝[set.Iio b] b) l
@[simp]
theorem
tendsto_comp_coe_Ioo_at_bot
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
{a b : α}
{l : filter β}
{f : α → β}
(h : a < b) :
filter.tendsto (λ (x : ↥(set.Ioo a b)), f ↑x) filter.at_bot l ↔ filter.tendsto f (𝓝[set.Ioi a] a) l
@[simp]
theorem
tendsto_comp_coe_Ioi_at_bot
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
{a : α}
{l : filter β}
{f : α → β} :
filter.tendsto (λ (x : ↥(set.Ioi a)), f ↑x) filter.at_bot l ↔ filter.tendsto f (𝓝[set.Ioi a] a) l
@[simp]
theorem
tendsto_comp_coe_Iio_at_top
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
{a : α}
{l : filter β}
{f : α → β} :
filter.tendsto (λ (x : ↥(set.Iio a)), f ↑x) filter.at_top l ↔ filter.tendsto f (𝓝[set.Iio a] a) l
@[simp]
theorem
tendsto_Ioo_at_top
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
{a b : α}
{l : filter β}
{f : β → ↥(set.Ioo a b)} :
filter.tendsto f l filter.at_top ↔ filter.tendsto (λ (x : β), ↑(f x)) l (𝓝[set.Iio b] b)
@[simp]
theorem
tendsto_Ioo_at_bot
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
{a b : α}
{l : filter β}
{f : β → ↥(set.Ioo a b)} :
filter.tendsto f l filter.at_bot ↔ filter.tendsto (λ (x : β), ↑(f x)) l (𝓝[set.Ioi a] a)
@[simp]
theorem
tendsto_Ioi_at_bot
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
{a : α}
{l : filter β}
{f : β → ↥(set.Ioi a)} :
filter.tendsto f l filter.at_bot ↔ filter.tendsto (λ (x : β), ↑(f x)) l (𝓝[set.Ioi a] a)
@[simp]
theorem
tendsto_Iio_at_top
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[order_topology α]
[densely_ordered α]
{a : α}
{l : filter β}
{f : β → ↥(set.Iio a)} :
filter.tendsto f l filter.at_top ↔ filter.tendsto (λ (x : β), ↑(f x)) l (𝓝[set.Iio a] a)
theorem
Sup_mem_closure
{α : Type u}
[topological_space α]
[complete_linear_order α]
[order_topology α]
{s : set α}
(hs : s.nonempty) :
theorem
Inf_mem_closure
{α : Type u}
[topological_space α]
[complete_linear_order α]
[order_topology α]
{s : set α}
(hs : s.nonempty) :
theorem
is_closed.Sup_mem
{α : Type u}
[topological_space α]
[complete_linear_order α]
[order_topology α]
{s : set α}
(hs : s.nonempty)
(hc : is_closed s) :
theorem
is_closed.Inf_mem
{α : Type u}
[topological_space α]
[complete_linear_order α]
[order_topology α]
{s : set α}
(hs : s.nonempty)
(hc : is_closed s) :
theorem
map_Sup_of_continuous_at_of_monotone'
{α : Type u}
{β : Type v}
[complete_linear_order α]
[topological_space α]
[order_topology α]
[complete_linear_order β]
[topological_space β]
[order_topology β]
{f : α → β}
{s : set α}
(Cf : continuous_at f (Sup s))
(Mf : monotone f)
(hs : s.nonempty) :
A monotone function continuous at the supremum of a nonempty set sends this supremum to the supremum of the image of this set.
theorem
map_Sup_of_continuous_at_of_monotone
{α : Type u}
{β : Type v}
[complete_linear_order α]
[topological_space α]
[order_topology α]
[complete_linear_order β]
[topological_space β]
[order_topology β]
{f : α → β}
{s : set α}
(Cf : continuous_at f (Sup s))
(Mf : monotone f)
(fbot : f ⊥ = ⊥) :
A monotone function s sending bot to bot and continuous at the supremum of a set sends
this supremum to the supremum of the image of this set.
theorem
map_supr_of_continuous_at_of_monotone'
{α : Type u}
{β : Type v}
[complete_linear_order α]
[topological_space α]
[order_topology α]
[complete_linear_order β]
[topological_space β]
[order_topology β]
{ι : Sort u_1}
[nonempty ι]
{f : α → β}
{g : ι → α}
(Cf : continuous_at f (supr g))
(Mf : monotone f) :
f (⨆ (i : ι), g i) = ⨆ (i : ι), f (g i)
A monotone function continuous at the indexed supremum over a nonempty Sort sends this indexed
supremum to the indexed supremum of the composition.
theorem
map_supr_of_continuous_at_of_monotone
{α : Type u}
{β : Type v}
[complete_linear_order α]
[topological_space α]
[order_topology α]
[complete_linear_order β]
[topological_space β]
[order_topology β]
{ι : Sort u_1}
{f : α → β}
{g : ι → α}
(Cf : continuous_at f (supr g))
(Mf : monotone f)
(fbot : f ⊥ = ⊥) :
f (⨆ (i : ι), g i) = ⨆ (i : ι), f (g i)
If a monotone function sending bot to bot is continuous at the indexed supremum over
a Sort, then it sends this indexed supremum to the indexed supremum of the composition.
theorem
map_Inf_of_continuous_at_of_monotone'
{α : Type u}
{β : Type v}
[complete_linear_order α]
[topological_space α]
[order_topology α]
[complete_linear_order β]
[topological_space β]
[order_topology β]
{f : α → β}
{s : set α}
(Cf : continuous_at f (Inf s))
(Mf : monotone f)
(hs : s.nonempty) :
A monotone function continuous at the infimum of a nonempty set sends this infimum to the infimum of the image of this set.
theorem
map_Inf_of_continuous_at_of_monotone
{α : Type u}
{β : Type v}
[complete_linear_order α]
[topological_space α]
[order_topology α]
[complete_linear_order β]
[topological_space β]
[order_topology β]
{f : α → β}
{s : set α}
(Cf : continuous_at f (Inf s))
(Mf : monotone f)
(ftop : f ⊤ = ⊤) :
A monotone function s sending top to top and continuous at the infimum of a set sends
this infimum to the infimum of the image of this set.
theorem
map_infi_of_continuous_at_of_monotone'
{α : Type u}
{β : Type v}
[complete_linear_order α]
[topological_space α]
[order_topology α]
[complete_linear_order β]
[topological_space β]
[order_topology β]
{ι : Sort u_1}
[nonempty ι]
{f : α → β}
{g : ι → α}
(Cf : continuous_at f (infi g))
(Mf : monotone f) :
f (⨅ (i : ι), g i) = ⨅ (i : ι), f (g i)
A monotone function continuous at the indexed infimum over a nonempty Sort sends this indexed
infimum to the indexed infimum of the composition.
theorem
map_infi_of_continuous_at_of_monotone
{α : Type u}
{β : Type v}
[complete_linear_order α]
[topological_space α]
[order_topology α]
[complete_linear_order β]
[topological_space β]
[order_topology β]
{ι : Sort u_1}
{f : α → β}
{g : ι → α}
(Cf : continuous_at f (infi g))
(Mf : monotone f)
(ftop : f ⊤ = ⊤) :
If a monotone function sending top to top is continuous at the indexed infimum over
a Sort, then it sends this indexed infimum to the indexed infimum of the composition.
theorem
cSup_mem_closure
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{s : set α}
(hs : s.nonempty)
(B : bdd_above s) :
theorem
cInf_mem_closure
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{s : set α}
(hs : s.nonempty)
(B : bdd_below s) :
theorem
is_closed.cSup_mem
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{s : set α}
(hc : is_closed s)
(hs : s.nonempty)
(B : bdd_above s) :
theorem
is_closed.cInf_mem
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{s : set α}
(hc : is_closed s)
(hs : s.nonempty)
(B : bdd_below s) :
theorem
map_cSup_of_continuous_at_of_monotone
{α : Type u}
{β : Type v}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[conditionally_complete_linear_order β]
[topological_space β]
[order_topology β]
{f : α → β}
{s : set α}
(Cf : continuous_at f (Sup s))
(Mf : monotone f)
(ne : s.nonempty)
(H : bdd_above s) :
If a monotone function is continuous at the supremum of a nonempty bounded above set s,
then it sends this supremum to the supremum of the image of s.
theorem
map_csupr_of_continuous_at_of_monotone
{α : Type u}
{β : Type v}
{γ : Type w}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[conditionally_complete_linear_order β]
[topological_space β]
[order_topology β]
[nonempty γ]
{f : α → β}
{g : γ → α}
(Cf : continuous_at f (⨆ (i : γ), g i))
(Mf : monotone f)
(H : bdd_above (set.range g)) :
f (⨆ (i : γ), g i) = ⨆ (i : γ), f (g i)
If a monotone function is continuous at the indexed supremum of a bounded function on
a nonempty Sort, then it sends this supremum to the supremum of the composition.
theorem
map_cInf_of_continuous_at_of_monotone
{α : Type u}
{β : Type v}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[conditionally_complete_linear_order β]
[topological_space β]
[order_topology β]
{f : α → β}
{s : set α}
(Cf : continuous_at f (Inf s))
(Mf : monotone f)
(ne : s.nonempty)
(H : bdd_below s) :
If a monotone function is continuous at the infimum of a nonempty bounded below set s,
then it sends this infimum to the infimum of the image of s.
theorem
map_cinfi_of_continuous_at_of_monotone
{α : Type u}
{β : Type v}
{γ : Type w}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[conditionally_complete_linear_order β]
[topological_space β]
[order_topology β]
[nonempty γ]
{f : α → β}
{g : γ → α}
(Cf : continuous_at f (⨅ (i : γ), g i))
(Mf : monotone f)
(H : bdd_below (set.range g)) :
f (⨅ (i : γ), g i) = ⨅ (i : γ), f (g i)
A continuous monotone function sends indexed infimum to indexed infimum in conditionally complete linear order, under a boundedness assumption.
theorem
is_connected.Ioo_cInf_cSup_subset
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{s : set α}
(hs : is_connected s)
(hb : bdd_below s)
(ha : bdd_above s) :
A bounded connected subset of a conditionally complete linear order includes the open interval
(Inf s, Sup s).
theorem
eq_Icc_cInf_cSup_of_connected_bdd_closed
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{s : set α}
(hc : is_connected s)
(hb : bdd_below s)
(ha : bdd_above s)
(hcl : is_closed s) :
theorem
is_preconnected.Ioi_cInf_subset
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{s : set α}
(hs : is_preconnected s)
(hb : bdd_below s)
(ha : ¬bdd_above s) :
theorem
is_preconnected.Iio_cSup_subset
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{s : set α}
(hs : is_preconnected s)
(hb : ¬bdd_below s)
(ha : bdd_above s) :
theorem
is_preconnected.mem_intervals
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{s : set α}
(hs : is_preconnected s) :
A preconnected set in a conditionally complete linear order is either one of the intervals
[Inf s, Sup s], [Inf s, Sup s), (Inf s, Sup s], (Inf s, Sup s), [Inf s, +∞),
(Inf s, +∞), (-∞, Sup s], (-∞, Sup s), (-∞, +∞), or ∅. The converse statement requires
α to be densely ordererd.
theorem
set_of_is_preconnected_subset_of_ordered
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α] :
A preconnected set is either one of the intervals Icc, Ico, Ioc, Ioo, Ici, Ioi,
Iic, Iio, or univ, or ∅. The converse statement requires α to be densely ordererd. Though
one can represent ∅ as (Inf s, Inf s), we include it into the list of possible cases to improve
readability.
theorem
is_closed.mem_of_ge_of_forall_exists_gt
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{a b : α}
{s : set α}
(hs : is_closed (s ∩ set.Icc a b))
(ha : a ∈ s)
(hab : a ≤ b)
(hgt : ∀ (x : α), x ∈ s ∩ set.Ico a b → (s ∩ set.Ioc x b).nonempty) :
b ∈ s
A "continuous induction principle" for a closed interval: if a set s meets [a, b]
on a closed subset, contains a, and the set s ∩ [a, b) has no maximal point, then b ∈ s.
theorem
is_closed.Icc_subset_of_forall_exists_gt
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{a b : α}
{s : set α}
(hs : is_closed (s ∩ set.Icc a b))
(ha : a ∈ s)
(hgt : ∀ (x : α), x ∈ s ∩ set.Ico a b → ∀ (y : α), y ∈ set.Ioi x → (s ∩ set.Ioc x y).nonempty) :
A "continuous induction principle" for a closed interval: if a set s meets [a, b]
on a closed subset, contains a, and for any a ≤ x < y ≤ b, x ∈ s, the set s ∩ (x, y]
is not empty, then [a, b] ⊆ s.
theorem
is_closed.Icc_subset_of_forall_mem_nhds_within
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[densely_ordered α]
{a b : α}
{s : set α}
(hs : is_closed (s ∩ set.Icc a b))
(ha : a ∈ s)
(hgt : ∀ (x : α), x ∈ s ∩ set.Ico a b → s ∈ 𝓝[set.Ioi x] x) :
A "continuous induction principle" for a closed interval: if a set s meets [a, b]
on a closed subset, contains a, and for any x ∈ s ∩ [a, b) the set s includes some open
neighborhood of x within (x, +∞), then [a, b] ⊆ s.
theorem
is_preconnected_Icc
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[densely_ordered α]
{a b : α} :
is_preconnected (set.Icc a b)
A closed interval in a densely ordered conditionally complete linear order is preconnected.
theorem
is_preconnected_interval
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[densely_ordered α]
{a b : α} :
is_preconnected (set.interval a b)
theorem
is_preconnected_iff_ord_connected
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[densely_ordered α]
{s : set α} :
theorem
is_preconnected.ord_connected
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[densely_ordered α]
{s : set α} :
is_preconnected s → s.ord_connected
Alias of is_preconnected_iff_ord_connected.
theorem
set.ord_connected.is_preconnected
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[densely_ordered α]
{s : set α} :
s.ord_connected → is_preconnected s
Alias of is_preconnected_iff_ord_connected.
theorem
is_preconnected_Ici
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[densely_ordered α]
{a : α} :
theorem
is_preconnected_Iic
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[densely_ordered α]
{a : α} :
theorem
is_preconnected_Iio
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[densely_ordered α]
{a : α} :
theorem
is_preconnected_Ioi
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[densely_ordered α]
{a : α} :
theorem
is_preconnected_Ioo
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[densely_ordered α]
{a b : α} :
is_preconnected (set.Ioo a b)
theorem
is_preconnected_Ioc
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[densely_ordered α]
{a b : α} :
is_preconnected (set.Ioc a b)
theorem
is_preconnected_Ico
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[densely_ordered α]
{a b : α} :
is_preconnected (set.Ico a b)
@[protected, instance]
def
ordered_connected_space
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[densely_ordered α] :
theorem
set_of_is_preconnected_eq_of_ordered
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[densely_ordered α] :
In a dense conditionally complete linear order, the set of preconnected sets is exactly
the set of the intervals Icc, Ico, Ioc, Ioo, Ici, Ioi, Iic, Iio, (-∞, +∞),
or ∅. Though one can represent ∅ as (Inf s, Inf s), we include it into the list of
possible cases to improve readability.
theorem
intermediate_value_Icc
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[densely_ordered α]
{δ : Type u_1}
[linear_order δ]
[topological_space δ]
[order_closed_topology δ]
{a b : α}
(hab : a ≤ b)
{f : α → δ}
(hf : continuous_on f (set.Icc a b)) :
Intermediate Value Theorem for continuous functions on closed intervals, case f a ≤ t ≤ f b.
theorem
intermediate_value_Icc'
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[densely_ordered α]
{δ : Type u_1}
[linear_order δ]
[topological_space δ]
[order_closed_topology δ]
{a b : α}
(hab : a ≤ b)
{f : α → δ}
(hf : continuous_on f (set.Icc a b)) :
Intermediate Value Theorem for continuous functions on closed intervals, case f a ≥ t ≥ f b.
theorem
intermediate_value_interval
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[densely_ordered α]
{δ : Type u_1}
[linear_order δ]
[topological_space δ]
[order_closed_topology δ]
{a b : α}
{f : α → δ}
(hf : continuous_on f (set.interval a b)) :
set.interval (f a) (f b) ⊆ f '' set.interval a b
Intermediate Value Theorem for continuous functions on closed intervals, unordered case.
theorem
intermediate_value_Ico
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[densely_ordered α]
{δ : Type u_1}
[linear_order δ]
[topological_space δ]
[order_closed_topology δ]
{a b : α}
(hab : a ≤ b)
{f : α → δ}
(hf : continuous_on f (set.Icc a b)) :
theorem
intermediate_value_Ico'
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[densely_ordered α]
{δ : Type u_1}
[linear_order δ]
[topological_space δ]
[order_closed_topology δ]
{a b : α}
(hab : a ≤ b)
{f : α → δ}
(hf : continuous_on f (set.Icc a b)) :
theorem
intermediate_value_Ioc
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[densely_ordered α]
{δ : Type u_1}
[linear_order δ]
[topological_space δ]
[order_closed_topology δ]
{a b : α}
(hab : a ≤ b)
{f : α → δ}
(hf : continuous_on f (set.Icc a b)) :
theorem
intermediate_value_Ioc'
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[densely_ordered α]
{δ : Type u_1}
[linear_order δ]
[topological_space δ]
[order_closed_topology δ]
{a b : α}
(hab : a ≤ b)
{f : α → δ}
(hf : continuous_on f (set.Icc a b)) :
theorem
intermediate_value_Ioo
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[densely_ordered α]
{δ : Type u_1}
[linear_order δ]
[topological_space δ]
[order_closed_topology δ]
{a b : α}
(hab : a ≤ b)
{f : α → δ}
(hf : continuous_on f (set.Icc a b)) :
theorem
intermediate_value_Ioo'
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[densely_ordered α]
{δ : Type u_1}
[linear_order δ]
[topological_space δ]
[order_closed_topology δ]
{a b : α}
(hab : a ≤ b)
{f : α → δ}
(hf : continuous_on f (set.Icc a b)) :
theorem
continuous.surjective
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[densely_ordered α]
{δ : Type u_1}
[linear_order δ]
[topological_space δ]
[order_closed_topology δ]
{f : α → δ}
(hf : continuous f)
(h_top : filter.tendsto f filter.at_top filter.at_top)
(h_bot : filter.tendsto f filter.at_bot filter.at_bot) :
A continuous function which tendsto at_top at_top and to at_bot at_bot is surjective.
theorem
continuous.surjective'
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[densely_ordered α]
{δ : Type u_1}
[linear_order δ]
[topological_space δ]
[order_closed_topology δ]
{f : α → δ}
(hf : continuous f)
(h_top : filter.tendsto f filter.at_bot filter.at_top)
(h_bot : filter.tendsto f filter.at_top filter.at_bot) :
A continuous function which tendsto at_bot at_top and to at_top at_bot is surjective.
theorem
continuous_on.surj_on_of_tendsto
{α : Type u}
{β : Type v}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[conditionally_complete_linear_order β]
[topological_space β]
[order_topology β]
[densely_ordered α]
{f : α → β}
{s : set α}
[s.ord_connected]
(hs : s.nonempty)
(hf : continuous_on f s)
(hbot : filter.tendsto (λ (x : ↥s), f ↑x) filter.at_bot filter.at_bot)
(htop : filter.tendsto (λ (x : ↥s), f ↑x) filter.at_top filter.at_top) :
set.surj_on f s set.univ
If a function f : α → β is continuous on a nonempty interval s, its restriction to s
tends to at_bot : filter β along at_bot : filter ↥s and tends to at_top : filter β along
at_top : filter ↥s, then the restriction of f to s is surjective. We formulate the
conclusion as surj_on f s univ.
theorem
continuous_on.surj_on_of_tendsto'
{α : Type u}
{β : Type v}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
[conditionally_complete_linear_order β]
[topological_space β]
[order_topology β]
[densely_ordered α]
{f : α → β}
{s : set α}
[s.ord_connected]
(hs : s.nonempty)
(hf : continuous_on f s)
(hbot : filter.tendsto (λ (x : ↥s), f ↑x) filter.at_bot filter.at_top)
(htop : filter.tendsto (λ (x : ↥s), f ↑x) filter.at_top filter.at_bot) :
set.surj_on f s set.univ
If a function f : α → β is continuous on a nonempty interval s, its restriction to s
tends to at_top : filter β along at_bot : filter ↥s and tends to at_bot : filter β along
at_top : filter ↥s, then the restriction of f to s is surjective. We formulate the
conclusion as surj_on f s univ.
theorem
compact_Icc
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{a b : α} :
is_compact (set.Icc a b)
A closed interval in a conditionally complete linear order is compact.
theorem
compact_interval
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{a b : α} :
is_compact (set.interval a b)
An unordered closed interval in a conditionally complete linear order is compact.
theorem
compact_pi_Icc
{ι : Type u_1}
{α : ι → Type u_2}
[Π (i : ι), conditionally_complete_linear_order (α i)]
[Π (i : ι), topological_space (α i)]
[∀ (i : ι), order_topology (α i)]
(a b : Π (i : ι), α i) :
is_compact (set.Icc a b)
@[protected, instance]
def
compact_space_Icc
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
(a b : α) :
compact_space ↥(set.Icc a b)
@[protected, instance]
def
compact_space_pi_Icc
{ι : Type u_1}
{α : ι → Type u_2}
[Π (i : ι), conditionally_complete_linear_order (α i)]
[Π (i : ι), topological_space (α i)]
[∀ (i : ι), order_topology (α i)]
(a b : Π (i : ι), α i) :
compact_space ↥(set.Icc a b)
@[protected, instance]
def
compact_space_of_complete_linear_order
{α : Type u_1}
[complete_linear_order α]
[topological_space α]
[order_topology α] :
theorem
is_compact.Inf_mem
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{s : set α}
(hs : is_compact s)
(ne_s : s.nonempty) :
theorem
is_compact.Sup_mem
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{s : set α}
(hs : is_compact s)
(ne_s : s.nonempty) :
theorem
is_compact.is_glb_Inf
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{s : set α}
(hs : is_compact s)
(ne_s : s.nonempty) :
theorem
is_compact.is_lub_Sup
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{s : set α}
(hs : is_compact s)
(ne_s : s.nonempty) :
theorem
is_compact.is_least_Inf
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{s : set α}
(hs : is_compact s)
(ne_s : s.nonempty) :
theorem
is_compact.is_greatest_Sup
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{s : set α}
(hs : is_compact s)
(ne_s : s.nonempty) :
is_greatest s (Sup s)
theorem
is_compact.exists_is_least
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{s : set α}
(hs : is_compact s)
(ne_s : s.nonempty) :
∃ (x : α), is_least s x
theorem
is_compact.exists_is_greatest
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{s : set α}
(hs : is_compact s)
(ne_s : s.nonempty) :
∃ (x : α), is_greatest s x
theorem
is_compact.exists_is_glb
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{s : set α}
(hs : is_compact s)
(ne_s : s.nonempty) :
theorem
is_compact.exists_is_lub
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{s : set α}
(hs : is_compact s)
(ne_s : s.nonempty) :
theorem
is_compact.exists_Inf_image_eq
{β : Type v}
[conditionally_complete_linear_order β]
[topological_space β]
[order_topology β]
{α : Type u}
[topological_space α]
{s : set α}
(hs : is_compact s)
(ne_s : s.nonempty)
{f : α → β}
(hf : continuous_on f s) :
theorem
is_compact.exists_Sup_image_eq
{β : Type v}
[conditionally_complete_linear_order β]
[topological_space β]
[order_topology β]
{α : Type u}
[topological_space α]
{s : set α} :
is_compact s → s.nonempty → ∀ {f : α → β}, continuous_on f s → (∃ (x : α) (H : x ∈ s), Sup (f '' s) = f x)
theorem
eq_Icc_of_connected_compact
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{s : set α}
(h₁ : is_connected s)
(h₂ : is_compact s) :
theorem
is_compact.exists_forall_le
{β : Type v}
[conditionally_complete_linear_order β]
[topological_space β]
[order_topology β]
{α : Type u}
[topological_space α]
{s : set α}
(hs : is_compact s)
(ne_s : s.nonempty)
{f : α → β}
(hf : continuous_on f s) :
The extreme value theorem: a continuous function realizes its minimum on a compact set
theorem
is_compact.exists_forall_ge
{β : Type v}
[conditionally_complete_linear_order β]
[topological_space β]
[order_topology β]
{α : Type u}
[topological_space α]
{s : set α} :
is_compact s → s.nonempty → ∀ {f : α → β}, continuous_on f s → (∃ (x : α) (H : x ∈ s), ∀ (y : α), y ∈ s → f y ≤ f x)
The extreme value theorem: a continuous function realizes its maximum on a compact set
theorem
continuous.exists_forall_le
{β : Type v}
[conditionally_complete_linear_order β]
[topological_space β]
[order_topology β]
{α : Type u_1}
[topological_space α]
[nonempty α]
{f : α → β}
(hf : continuous f)
(hlim : filter.tendsto f (filter.cocompact α) filter.at_top) :
∃ (x : α), ∀ (y : α), f x ≤ f y
The extreme value theorem: if a continuous function f tends to infinity away from compact
sets, then it has a global minimum.
theorem
continuous.exists_forall_ge
{β : Type v}
[conditionally_complete_linear_order β]
[topological_space β]
[order_topology β]
{α : Type u_1}
[topological_space α]
[nonempty α]
{f : α → β}
(hf : continuous f)
(hlim : filter.tendsto f (filter.cocompact α) filter.at_bot) :
∃ (x : α), ∀ (y : α), f y ≤ f x
The extreme value theorem: if a continuous function f tends to negative infinity away from
compactx sets, then it has a global maximum.
theorem
is_bounded_le_nhds
{α : Type u}
[semilattice_sup α]
[topological_space α]
[order_topology α]
(a : α) :
theorem
filter.tendsto.is_bounded_under_le
{α : Type u}
{β : Type v}
[semilattice_sup α]
[topological_space α]
[order_topology α]
{f : filter β}
{u : β → α}
{a : α}
(h : filter.tendsto u f (𝓝 a)) :
theorem
is_cobounded_ge_nhds
{α : Type u}
[semilattice_sup α]
[topological_space α]
[order_topology α]
(a : α) :
filter.is_cobounded ge (𝓝 a)
theorem
filter.tendsto.is_cobounded_under_ge
{α : Type u}
{β : Type v}
[semilattice_sup α]
[topological_space α]
[order_topology α]
{f : filter β}
{u : β → α}
{a : α}
[f.ne_bot]
(h : filter.tendsto u f (𝓝 a)) :
theorem
is_bounded_ge_nhds
{α : Type u}
[semilattice_inf α]
[topological_space α]
[order_topology α]
(a : α) :
filter.is_bounded ge (𝓝 a)
theorem
filter.tendsto.is_bounded_under_ge
{α : Type u}
{β : Type v}
[semilattice_inf α]
[topological_space α]
[order_topology α]
{f : filter β}
{u : β → α}
{a : α}
(h : filter.tendsto u f (𝓝 a)) :
theorem
is_cobounded_le_nhds
{α : Type u}
[semilattice_inf α]
[topological_space α]
[order_topology α]
(a : α) :
theorem
filter.tendsto.is_cobounded_under_le
{α : Type u}
{β : Type v}
[semilattice_inf α]
[topological_space α]
[order_topology α]
{f : filter β}
{u : β → α}
{a : α}
[f.ne_bot]
(h : filter.tendsto u f (𝓝 a)) :
theorem
lt_mem_sets_of_Limsup_lt
{α : Type u}
[conditionally_complete_linear_order α]
{f : filter α}
{b : α}
(h : filter.is_bounded has_le.le f)
(l : f.Limsup < b) :
∀ᶠ (a : α) in f, a < b
theorem
gt_mem_sets_of_Liminf_gt
{α : Type u}
[conditionally_complete_linear_order α]
{f : filter α}
{b : α} :
filter.is_bounded ge f → b < f.Liminf → (∀ᶠ (a : α) in f, b < a)
theorem
le_nhds_of_Limsup_eq_Liminf
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{f : filter α}
{a : α}
(hl : filter.is_bounded has_le.le f)
(hg : filter.is_bounded ge f)
(hs : f.Limsup = a)
(hi : f.Liminf = a) :
If the liminf and the limsup of a filter coincide, then this filter converges to their common value, at least if the filter is eventually bounded above and below.
theorem
Limsup_nhds
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
(a : α) :
theorem
Liminf_nhds
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
(a : α) :
theorem
Liminf_eq_of_le_nhds
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{f : filter α}
{a : α}
[f.ne_bot]
(h : f ≤ 𝓝 a) :
If a filter is converging, its limsup coincides with its limit.
theorem
Limsup_eq_of_le_nhds
{α : Type u}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{f : filter α}
{a : α}
[f.ne_bot] :
If a filter is converging, its liminf coincides with its limit.
theorem
filter.tendsto.limsup_eq
{α : Type u}
{β : Type v}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{f : filter β}
{u : β → α}
{a : α}
[f.ne_bot]
(h : filter.tendsto u f (𝓝 a)) :
If a function has a limit, then its limsup coincides with its limit.
theorem
filter.tendsto.liminf_eq
{α : Type u}
{β : Type v}
[conditionally_complete_linear_order α]
[topological_space α]
[order_topology α]
{f : filter β}
{u : β → α}
{a : α}
[f.ne_bot]
(h : filter.tendsto u f (𝓝 a)) :
If a function has a limit, then its liminf coincides with its limit.
theorem
tendsto_of_liminf_eq_limsup
{α : Type u}
{β : Type v}
[complete_linear_order α]
[topological_space α]
[order_topology α]
{f : filter β}
{u : β → α}
{a : α}
(hinf : f.liminf u = a)
(hsup : f.limsup u = a) :
filter.tendsto u f (𝓝 a)
If the liminf and the limsup of a function coincide, then the limit of the function exists and has the same value
theorem
tendsto_of_le_liminf_of_limsup_le
{α : Type u}
{β : Type v}
[complete_linear_order α]
[topological_space α]
[order_topology α]
{f : filter β}
{u : β → α}
{a : α}
(hinf : a ≤ f.liminf u)
(hsup : f.limsup u ≤ a) :
filter.tendsto u f (𝓝 a)
If a number a is less than or equal to the liminf of a function f at some filter
and is greater than or equal to the limsup of f, then f tends to a along this filter.
Here is a counter-example to a version of the following with conditionally_complete_lattice α.
Take α = [0, 1) → ℝ with the natural lattice structure, ι = ℕ. Put f n x = -x^n. Then
⨆ n, f n = 0 while none of f n is strictly greater than the constant function -0.5.
theorem
tendsto_at_top_csupr
{ι : Type u_1}
{α : Type u_2}
[preorder ι]
[topological_space α]
[conditionally_complete_linear_order α]
[order_topology α]
{f : ι → α}
(h_mono : monotone f)
(hbdd : bdd_above (set.range f)) :
filter.tendsto f filter.at_top (𝓝 (⨆ (i : ι), f i))
theorem
tendsto_at_bot_cinfi
{ι : Type u_1}
{α : Type u_2}
[preorder ι]
[topological_space α]
[conditionally_complete_linear_order α]
[order_topology α]
{f : ι → α}
(h_mono : monotone f)
(hbdd : bdd_below (set.range f)) :
filter.tendsto f filter.at_bot (𝓝 (⨅ (i : ι), f i))
theorem
tendsto_at_top_cinfi
{ι : Type u_1}
{α : Type u_2}
[preorder ι]
[topological_space α]
[conditionally_complete_linear_order α]
[order_topology α]
{f : ι → α}
(h_mono : ∀ ⦃i j : ι⦄, i ≤ j → f j ≤ f i)
(hbdd : bdd_below (set.range f)) :
filter.tendsto f filter.at_top (𝓝 (⨅ (i : ι), f i))
theorem
tendsto_at_bot_csupr
{ι : Type u_1}
{α : Type u_2}
[preorder ι]
[topological_space α]
[conditionally_complete_linear_order α]
[order_topology α]
{f : ι → α}
(h_mono : ∀ ⦃i j : ι⦄, i ≤ j → f j ≤ f i)
(hbdd : bdd_above (set.range f)) :
filter.tendsto f filter.at_bot (𝓝 (⨆ (i : ι), f i))
theorem
tendsto_at_top_supr
{ι : Type u_1}
{α : Type u_2}
[preorder ι]
[topological_space α]
[complete_linear_order α]
[order_topology α]
{f : ι → α}
(h_mono : monotone f) :
filter.tendsto f filter.at_top (𝓝 (⨆ (i : ι), f i))
theorem
tendsto_at_bot_infi
{ι : Type u_1}
{α : Type u_2}
[preorder ι]
[topological_space α]
[complete_linear_order α]
[order_topology α]
{f : ι → α}
(h_mono : monotone f) :
filter.tendsto f filter.at_bot (𝓝 (⨅ (i : ι), f i))
theorem
tendsto_at_top_infi
{ι : Type u_1}
{α : Type u_2}
[preorder ι]
[topological_space α]
[complete_linear_order α]
[order_topology α]
{f : ι → α}
(h_mono : ∀ ⦃i j : ι⦄, i ≤ j → f j ≤ f i) :
filter.tendsto f filter.at_top (𝓝 (⨅ (i : ι), f i))
theorem
tendsto_at_bot_supr
{ι : Type u_1}
{α : Type u_2}
[preorder ι]
[topological_space α]
[complete_linear_order α]
[order_topology α]
{f : ι → α}
(h_mono : ∀ ⦃i j : ι⦄, i ≤ j → f j ≤ f i) :
filter.tendsto f filter.at_bot (𝓝 (⨆ (i : ι), f i))
theorem
tendsto_of_monotone
{ι : Type u_1}
{α : Type u_2}
[preorder ι]
[topological_space α]
[conditionally_complete_linear_order α]
[order_topology α]
{f : ι → α}
(h_mono : monotone f) :
filter.tendsto f filter.at_top filter.at_top ∨ ∃ (l : α), filter.tendsto f filter.at_top (𝓝 l)
theorem
supr_eq_of_tendsto
{α : Type u_1}
{β : Type u_2}
[topological_space α]
[complete_linear_order α]
[order_topology α]
[nonempty β]
[semilattice_sup β]
{f : β → α}
{a : α}
(hf : monotone f) :
filter.tendsto f filter.at_top (𝓝 a) → supr f = a
theorem
infi_eq_of_tendsto
{β : Type v}
{α : Type u_1}
[topological_space α]
[complete_linear_order α]
[order_topology α]
[nonempty β]
[semilattice_sup β]
{f : β → α}
{a : α}
(hf : ∀ (n m : β), n ≤ m → f m ≤ f n) :
filter.tendsto f filter.at_top (𝓝 a) → infi f = a
theorem
tendsto_inv_nhds_within_Ioi
{α : Type u}
[ordered_comm_group α]
[topological_space α]
[topological_group α]
{a : α} :
filter.tendsto has_inv.inv (𝓝[set.Ioi a] a) (𝓝[set.Iio a⁻¹] a⁻¹)
theorem
tendsto_neg_nhds_within_Ioi
{α : Type u}
[ordered_add_comm_group α]
[topological_space α]
[topological_add_group α]
{a : α} :
filter.tendsto has_neg.neg (𝓝[set.Ioi a] a) (𝓝[set.Iio (-a)] -a)
theorem
tendsto_inv_nhds_within_Iio
{α : Type u}
[ordered_comm_group α]
[topological_space α]
[topological_group α]
{a : α} :
filter.tendsto has_inv.inv (𝓝[set.Iio a] a) (𝓝[set.Ioi a⁻¹] a⁻¹)
theorem
tendsto_neg_nhds_within_Iio
{α : Type u}
[ordered_add_comm_group α]
[topological_space α]
[topological_add_group α]
{a : α} :
filter.tendsto has_neg.neg (𝓝[set.Iio a] a) (𝓝[set.Ioi (-a)] -a)
theorem
tendsto_inv_nhds_within_Ioi_inv
{α : Type u}
[ordered_comm_group α]
[topological_space α]
[topological_group α]
{a : α} :
filter.tendsto has_inv.inv (𝓝[set.Ioi a⁻¹] a⁻¹) (𝓝[set.Iio a] a)
theorem
tendsto_neg_nhds_within_Ioi_neg
{α : Type u}
[ordered_add_comm_group α]
[topological_space α]
[topological_add_group α]
{a : α} :
filter.tendsto has_neg.neg (𝓝[set.Ioi (-a)] -a) (𝓝[set.Iio a] a)
theorem
tendsto_inv_nhds_within_Iio_inv
{α : Type u}
[ordered_comm_group α]
[topological_space α]
[topological_group α]
{a : α} :
filter.tendsto has_inv.inv (𝓝[set.Iio a⁻¹] a⁻¹) (𝓝[set.Ioi a] a)
theorem
tendsto_neg_nhds_within_Iio_neg
{α : Type u}
[ordered_add_comm_group α]
[topological_space α]
[topological_add_group α]
{a : α} :
filter.tendsto has_neg.neg (𝓝[set.Iio (-a)] -a) (𝓝[set.Ioi a] a)
theorem
tendsto_inv_nhds_within_Ici
{α : Type u}
[ordered_comm_group α]
[topological_space α]
[topological_group α]
{a : α} :
filter.tendsto has_inv.inv (𝓝[set.Ici a] a) (𝓝[set.Iic a⁻¹] a⁻¹)
theorem
tendsto_neg_nhds_within_Ici
{α : Type u}
[ordered_add_comm_group α]
[topological_space α]
[topological_add_group α]
{a : α} :
filter.tendsto has_neg.neg (𝓝[set.Ici a] a) (𝓝[set.Iic (-a)] -a)
theorem
tendsto_inv_nhds_within_Iic
{α : Type u}
[ordered_comm_group α]
[topological_space α]
[topological_group α]
{a : α} :
filter.tendsto has_inv.inv (𝓝[set.Iic a] a) (𝓝[set.Ici a⁻¹] a⁻¹)
theorem
tendsto_neg_nhds_within_Iic
{α : Type u}
[ordered_add_comm_group α]
[topological_space α]
[topological_add_group α]
{a : α} :
filter.tendsto has_neg.neg (𝓝[set.Iic a] a) (𝓝[set.Ici (-a)] -a)
theorem
tendsto_neg_nhds_within_Ici_neg
{α : Type u}
[ordered_add_comm_group α]
[topological_space α]
[topological_add_group α]
{a : α} :
filter.tendsto has_neg.neg (𝓝[set.Ici (-a)] -a) (𝓝[set.Iic a] a)
theorem
tendsto_inv_nhds_within_Ici_inv
{α : Type u}
[ordered_comm_group α]
[topological_space α]
[topological_group α]
{a : α} :
filter.tendsto has_inv.inv (𝓝[set.Ici a⁻¹] a⁻¹) (𝓝[set.Iic a] a)
theorem
tendsto_neg_nhds_within_Iic_neg
{α : Type u}
[ordered_add_comm_group α]
[topological_space α]
[topological_add_group α]
{a : α} :
filter.tendsto has_neg.neg (𝓝[set.Iic (-a)] -a) (𝓝[set.Ici a] a)
theorem
tendsto_inv_nhds_within_Iic_inv
{α : Type u}
[ordered_comm_group α]
[topological_space α]
[topological_group α]
{a : α} :
filter.tendsto has_inv.inv (𝓝[set.Iic a⁻¹] a⁻¹) (𝓝[set.Ici a] a)
theorem
continuous_at_iff_continuous_left_right
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[topological_space β]
{a : α}
{f : α → β} :
continuous_at f a ↔ continuous_within_at f (set.Iic a) a ∧ continuous_within_at f (set.Ici a) a
theorem
continuous_on_Icc_extend_from_Ioo
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[densely_ordered α]
[order_topology α]
[topological_space β]
[regular_space β]
{f : α → β}
{a b : α}
{la lb : β}
(hab : a < b)
(hf : continuous_on f (set.Ioo a b))
(ha : filter.tendsto f (𝓝[set.Ioi a] a) (𝓝 la))
(hb : filter.tendsto f (𝓝[set.Iio b] b) (𝓝 lb)) :
continuous_on (extend_from (set.Ioo a b) f) (set.Icc a b)
theorem
eq_lim_at_left_extend_from_Ioo
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[densely_ordered α]
[order_topology α]
[topological_space β]
[t2_space β]
{f : α → β}
{a b : α}
{la : β}
(hab : a < b)
(ha : filter.tendsto f (𝓝[set.Ioi a] a) (𝓝 la)) :
extend_from (set.Ioo a b) f a = la
theorem
eq_lim_at_right_extend_from_Ioo
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[densely_ordered α]
[order_topology α]
[topological_space β]
[t2_space β]
{f : α → β}
{a b : α}
{lb : β}
(hab : a < b)
(hb : filter.tendsto f (𝓝[set.Iio b] b) (𝓝 lb)) :
extend_from (set.Ioo a b) f b = lb
theorem
continuous_on_Ico_extend_from_Ioo
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[densely_ordered α]
[order_topology α]
[topological_space β]
[regular_space β]
{f : α → β}
{a b : α}
{la : β}
(hab : a < b)
(hf : continuous_on f (set.Ioo a b))
(ha : filter.tendsto f (𝓝[set.Ioi a] a) (𝓝 la)) :
continuous_on (extend_from (set.Ioo a b) f) (set.Ico a b)
theorem
continuous_on_Ioc_extend_from_Ioo
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[densely_ordered α]
[order_topology α]
[topological_space β]
[regular_space β]
{f : α → β}
{a b : α}
{lb : β}
(hab : a < b)
(hf : continuous_on f (set.Ioo a b))
(hb : filter.tendsto f (𝓝[set.Iio b] b) (𝓝 lb)) :
continuous_on (extend_from (set.Ioo a b) f) (set.Ioc a b)
theorem
continuous_within_at_Ioi_iff_Ici
{α : Type u_1}
{β : Type u_2}
[topological_space α]
[partial_order α]
[topological_space β]
{a : α}
{f : α → β} :
continuous_within_at f (set.Ioi a) a ↔ continuous_within_at f (set.Ici a) a
theorem
continuous_within_at_Iio_iff_Iic
{α : Type u_1}
{β : Type u_2}
[topological_space α]
[linear_order α]
[topological_space β]
{a : α}
{f : α → β} :
continuous_within_at f (set.Iio a) a ↔ continuous_within_at f (set.Iic a) a
theorem
continuous_at_iff_continuous_left'_right'
{α : Type u}
{β : Type v}
[topological_space α]
[linear_order α]
[topological_space β]
{a : α}
{f : α → β} :
continuous_at f a ↔ continuous_within_at f (set.Iio a) a ∧ continuous_within_at f (set.Ioi a) a
Continuity of monotone functions #
In this section we prove the following fact: if f is a monotone function on a neighborhood of a
and the image of this neighborhood is a neighborhood of f a, then f is continuous at a, see
continuous_at_of_mono_incr_on_of_image_mem_nhds, as well as several similar facts.
theorem
strict_mono_incr_on.continuous_at_right_of_exists_between
{α : Type u}
{β : Type v}
[linear_order α]
[topological_space α]
[order_topology α]
[linear_order β]
[topological_space β]
[order_topology β]
{f : α → β}
{s : set α}
{a : α}
(h_mono : strict_mono_incr_on f s)
(hs : s ∈ 𝓝[set.Ici a] a)
(hfs : ∀ (b : β), b > f a → (∃ (c : α) (H : c ∈ s), f c ∈ set.Ioc (f a) b)) :
continuous_within_at f (set.Ici a) a
If f is a function strictly monotonically increasing on a right neighborhood of a and the
image of this neighborhood under f meets every interval (f a, b], b > f a, then f is
continuous at a from the right.
The assumption hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b is required because otherwise the
function f : ℝ → ℝ given by f x = if x ≤ 0 then x else x + 1 would be a counter-example at
a = 0.
theorem
continuous_at_right_of_mono_incr_on_of_exists_between
{α : Type u}
{β : Type v}
[linear_order α]
[topological_space α]
[order_topology α]
[linear_order β]
[topological_space β]
[order_topology β]
{f : α → β}
{s : set α}
{a : α}
(h_mono : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → x ≤ y → f x ≤ f y)
(hs : s ∈ 𝓝[set.Ici a] a)
(hfs : ∀ (b : β), b > f a → (∃ (c : α) (H : c ∈ s), f c ∈ set.Ioo (f a) b)) :
continuous_within_at f (set.Ici a) a
If f is a function monotonically increasing function on a right neighborhood of a and the
image of this neighborhood under f meets every interval (f a, b), b > f a, then f is
continuous at a from the right.
The assumption hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioo (f a) b cannot be replaced by the weaker
assumption hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b we use for strictly monotone functions
because otherwise the function ceil : ℝ → ℤ would be a counter-example at a = 0.
theorem
continuous_at_right_of_mono_incr_on_of_closure_image_mem_nhds_within
{α : Type u}
{β : Type v}
[linear_order α]
[topological_space α]
[order_topology α]
[linear_order β]
[topological_space β]
[order_topology β]
[densely_ordered β]
{f : α → β}
{s : set α}
{a : α}
(h_mono : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → x ≤ y → f x ≤ f y)
(hs : s ∈ 𝓝[set.Ici a] a)
(hfs : closure (f '' s) ∈ 𝓝[set.Ici (f a)] f a) :
continuous_within_at f (set.Ici a) a
If a function f with a densely ordered codomain is monotonically increasing on a right
neighborhood of a and the closure of the image of this neighborhood under f is a right
neighborhood of f a, then f is continuous at a from the right.
theorem
continuous_at_right_of_mono_incr_on_of_image_mem_nhds_within
{α : Type u}
{β : Type v}
[linear_order α]
[topological_space α]
[order_topology α]
[linear_order β]
[topological_space β]
[order_topology β]
[densely_ordered β]
{f : α → β}
{s : set α}
{a : α}
(h_mono : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → x ≤ y → f x ≤ f y)
(hs : s ∈ 𝓝[set.Ici a] a)
(hfs : f '' s ∈ 𝓝[set.Ici (f a)] f a) :
continuous_within_at f (set.Ici a) a
If a function f with a densely ordered codomain is monotonically increasing on a right
neighborhood of a and the image of this neighborhood under f is a right neighborhood of f a,
then f is continuous at a from the right.
theorem
strict_mono_incr_on.continuous_at_right_of_closure_image_mem_nhds_within
{α : Type u}
{β : Type v}
[linear_order α]
[topological_space α]
[order_topology α]
[linear_order β]
[topological_space β]
[order_topology β]
[densely_ordered β]
{f : α → β}
{s : set α}
{a : α}
(h_mono : strict_mono_incr_on f s)
(hs : s ∈ 𝓝[set.Ici a] a)
(hfs : closure (f '' s) ∈ 𝓝[set.Ici (f a)] f a) :
continuous_within_at f (set.Ici a) a
If a function f with a densely ordered codomain is strictly monotonically increasing on a
right neighborhood of a and the closure of the image of this neighborhood under f is a right
neighborhood of f a, then f is continuous at a from the right.
theorem
strict_mono_incr_on.continuous_at_right_of_image_mem_nhds_within
{α : Type u}
{β : Type v}
[linear_order α]
[topological_space α]
[order_topology α]
[linear_order β]
[topological_space β]
[order_topology β]
[densely_ordered β]
{f : α → β}
{s : set α}
{a : α}
(h_mono : strict_mono_incr_on f s)
(hs : s ∈ 𝓝[set.Ici a] a)
(hfs : f '' s ∈ 𝓝[set.Ici (f a)] f a) :
continuous_within_at f (set.Ici a) a
If a function f with a densely ordered codomain is strictly monotonically increasing on a
right neighborhood of a and the image of this neighborhood under f is a right neighborhood of
f a, then f is continuous at a from the right.
theorem
strict_mono_incr_on.continuous_at_right_of_surj_on
{α : Type u}
{β : Type v}
[linear_order α]
[topological_space α]
[order_topology α]
[linear_order β]
[topological_space β]
[order_topology β]
{f : α → β}
{s : set α}
{a : α}
(h_mono : strict_mono_incr_on f s)
(hs : s ∈ 𝓝[set.Ici a] a)
(hfs : set.surj_on f s (set.Ioi (f a))) :
continuous_within_at f (set.Ici a) a
If a function f is strictly monotonically increasing on a right neighborhood of a and the
image of this neighborhood under f includes Ioi (f a), then f is continuous at a from the
right.
theorem
strict_mono_incr_on.continuous_at_left_of_exists_between
{α : Type u}
{β : Type v}
[linear_order α]
[topological_space α]
[order_topology α]
[linear_order β]
[topological_space β]
[order_topology β]
{f : α → β}
{s : set α}
{a : α}
(h_mono : strict_mono_incr_on f s)
(hs : s ∈ 𝓝[set.Iic a] a)
(hfs : ∀ (b : β), b < f a → (∃ (c : α) (H : c ∈ s), f c ∈ set.Ico b (f a))) :
continuous_within_at f (set.Iic a) a
If f is a function strictly monotonically increasing on a left neighborhood of a and the
image of this neighborhood under f meets every interval [b, f a), b < f a, then f is
continuous at a from the left.
The assumption hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ico b (f a) is required because otherwise the
function f : ℝ → ℝ given by f x = if x < 0 then x else x + 1 would be a counter-example at
a = 0.
theorem
continuous_at_left_of_mono_incr_on_of_exists_between
{α : Type u}
{β : Type v}
[linear_order α]
[topological_space α]
[order_topology α]
[linear_order β]
[topological_space β]
[order_topology β]
{f : α → β}
{s : set α}
{a : α}
(h_mono : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → x ≤ y → f x ≤ f y)
(hs : s ∈ 𝓝[set.Iic a] a)
(hfs : ∀ (b : β), b < f a → (∃ (c : α) (H : c ∈ s), f c ∈ set.Ioo b (f a))) :
continuous_within_at f (set.Iic a) a
If f is a function monotonically increasing function on a left neighborhood of a and the
image of this neighborhood under f meets every interval (b, f a), b < f a, then f is
continuous at a from the left.
The assumption hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ioo b (f a) cannot be replaced by the weaker
assumption hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ico b (f a) we use for strictly monotone functions
because otherwise the function floor : ℝ → ℤ would be a counter-example at a = 0.
theorem
continuous_at_left_of_mono_incr_on_of_closure_image_mem_nhds_within
{α : Type u}
{β : Type v}
[linear_order α]
[topological_space α]
[order_topology α]
[linear_order β]
[topological_space β]
[order_topology β]
[densely_ordered β]
{f : α → β}
{s : set α}
{a : α}
(h_mono : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → x ≤ y → f x ≤ f y)
(hs : s ∈ 𝓝[set.Iic a] a)
(hfs : closure (f '' s) ∈ 𝓝[set.Iic (f a)] f a) :
continuous_within_at f (set.Iic a) a
If a function f with a densely ordered codomain is monotonically increasing on a left
neighborhood of a and the closure of the image of this neighborhood under f is a left
neighborhood of f a, then f is continuous at a from the left
theorem
continuous_at_left_of_mono_incr_on_of_image_mem_nhds_within
{α : Type u}
{β : Type v}
[linear_order α]
[topological_space α]
[order_topology α]
[linear_order β]
[topological_space β]
[order_topology β]
[densely_ordered β]
{f : α → β}
{s : set α}
{a : α}
(h_mono : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → x ≤ y → f x ≤ f y)
(hs : s ∈ 𝓝[set.Iic a] a)
(hfs : f '' s ∈ 𝓝[set.Iic (f a)] f a) :
continuous_within_at f (set.Iic a) a
If a function f with a densely ordered codomain is monotonically increasing on a left
neighborhood of a and the image of this neighborhood under f is a left neighborhood of f a,
then f is continuous at a from the left.
theorem
strict_mono_incr_on.continuous_at_left_of_closure_image_mem_nhds_within
{α : Type u}
{β : Type v}
[linear_order α]
[topological_space α]
[order_topology α]
[linear_order β]
[topological_space β]
[order_topology β]
[densely_ordered β]
{f : α → β}
{s : set α}
{a : α}
(h_mono : strict_mono_incr_on f s)
(hs : s ∈ 𝓝[set.Iic a] a)
(hfs : closure (f '' s) ∈ 𝓝[set.Iic (f a)] f a) :
continuous_within_at f (set.Iic a) a
If a function f with a densely ordered codomain is strictly monotonically increasing on a
left neighborhood of a and the closure of the image of this neighborhood under f is a left
neighborhood of f a, then f is continuous at a from the left.
theorem
strict_mono_incr_on.continuous_at_left_of_image_mem_nhds_within
{α : Type u}
{β : Type v}
[linear_order α]
[topological_space α]
[order_topology α]
[linear_order β]
[topological_space β]
[order_topology β]
[densely_ordered β]
{f : α → β}
{s : set α}
{a : α}
(h_mono : strict_mono_incr_on f s)
(hs : s ∈ 𝓝[set.Iic a] a)
(hfs : f '' s ∈ 𝓝[set.Iic (f a)] f a) :
continuous_within_at f (set.Iic a) a
If a function f with a densely ordered codomain is strictly monotonically increasing on a
left neighborhood of a and the image of this neighborhood under f is a left neighborhood of
f a, then f is continuous at a from the left.
theorem
strict_mono_incr_on.continuous_at_left_of_surj_on
{α : Type u}
{β : Type v}
[linear_order α]
[topological_space α]
[order_topology α]
[linear_order β]
[topological_space β]
[order_topology β]
{f : α → β}
{s : set α}
{a : α}
(h_mono : strict_mono_incr_on f s)
(hs : s ∈ 𝓝[set.Iic a] a)
(hfs : set.surj_on f s (set.Iio (f a))) :
continuous_within_at f (set.Iic a) a
If a function f is strictly monotonically increasing on a left neighborhood of a and the
image of this neighborhood under f includes Iio (f a), then f is continuous at a from the
left.
theorem
strict_mono_incr_on.continuous_at_of_exists_between
{α : Type u}
{β : Type v}
[linear_order α]
[topological_space α]
[order_topology α]
[linear_order β]
[topological_space β]
[order_topology β]
{f : α → β}
{s : set α}
{a : α}
(h_mono : strict_mono_incr_on f s)
(hs : s ∈ 𝓝 a)
(hfs_l : ∀ (b : β), b < f a → (∃ (c : α) (H : c ∈ s), f c ∈ set.Ico b (f a)))
(hfs_r : ∀ (b : β), b > f a → (∃ (c : α) (H : c ∈ s), f c ∈ set.Ioc (f a) b)) :
continuous_at f a
If a function f is strictly monotonically increasing on a neighborhood of a and the image of
this neighborhood under f meets every interval [b, f a), b < f a, and every interval
(f a, b], b > f a, then f is continuous at a.
theorem
strict_mono_incr_on.continuous_at_of_closure_image_mem_nhds
{α : Type u}
{β : Type v}
[linear_order α]
[topological_space α]
[order_topology α]
[linear_order β]
[topological_space β]
[order_topology β]
[densely_ordered β]
{f : α → β}
{s : set α}
{a : α}
(h_mono : strict_mono_incr_on f s)
(hs : s ∈ 𝓝 a)
(hfs : closure (f '' s) ∈ 𝓝 (f a)) :
continuous_at f a
If a function f with a densely ordered codomain is strictly monotonically increasing on a
neighborhood of a and the closure of the image of this neighborhood under f is a neighborhood of
f a, then f is continuous at a.
theorem
strict_mono_incr_on.continuous_at_of_image_mem_nhds
{α : Type u}
{β : Type v}
[linear_order α]
[topological_space α]
[order_topology α]
[linear_order β]
[topological_space β]
[order_topology β]
[densely_ordered β]
{f : α → β}
{s : set α}
{a : α}
(h_mono : strict_mono_incr_on f s)
(hs : s ∈ 𝓝 a)
(hfs : f '' s ∈ 𝓝 (f a)) :
continuous_at f a
If a function f with a densely ordered codomain is strictly monotonically increasing on a
neighborhood of a and the image of this set under f is a neighborhood of f a, then f is
continuous at a.
theorem
continuous_at_of_mono_incr_on_of_exists_between
{α : Type u}
{β : Type v}
[linear_order α]
[topological_space α]
[order_topology α]
[linear_order β]
[topological_space β]
[order_topology β]
{f : α → β}
{s : set α}
{a : α}
(h_mono : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → x ≤ y → f x ≤ f y)
(hs : s ∈ 𝓝 a)
(hfs_l : ∀ (b : β), b < f a → (∃ (c : α) (H : c ∈ s), f c ∈ set.Ioo b (f a)))
(hfs_r : ∀ (b : β), b > f a → (∃ (c : α) (H : c ∈ s), f c ∈ set.Ioo (f a) b)) :
continuous_at f a
If f is a function monotonically increasing function on a neighborhood of a and the image of
this neighborhood under f meets every interval (b, f a), b < f a, and every interval (f a, b), b > f a, then f is continuous at a.
theorem
continuous_at_of_mono_incr_on_of_closure_image_mem_nhds
{α : Type u}
{β : Type v}
[linear_order α]
[topological_space α]
[order_topology α]
[linear_order β]
[topological_space β]
[order_topology β]
[densely_ordered β]
{f : α → β}
{s : set α}
{a : α}
(h_mono : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → x ≤ y → f x ≤ f y)
(hs : s ∈ 𝓝 a)
(hfs : closure (f '' s) ∈ 𝓝 (f a)) :
continuous_at f a
If a function f with a densely ordered codomain is monotonically increasing on a neighborhood
of a and the closure of the image of this neighborhood under f is a neighborhood of f a, then
f is continuous at a.
theorem
continuous_at_of_mono_incr_on_of_image_mem_nhds
{α : Type u}
{β : Type v}
[linear_order α]
[topological_space α]
[order_topology α]
[linear_order β]
[topological_space β]
[order_topology β]
[densely_ordered β]
{f : α → β}
{s : set α}
{a : α}
(h_mono : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → x ≤ y → f x ≤ f y)
(hs : s ∈ 𝓝 a)
(hfs : f '' s ∈ 𝓝 (f a)) :
continuous_at f a
If a function f with a densely ordered codomain is monotonically increasing on a neighborhood
of a and the image of this neighborhood under f is a neighborhood of f a, then f is
continuous at a.
theorem
monotone.continuous_of_dense_range
{α : Type u}
{β : Type v}
[linear_order α]
[topological_space α]
[order_topology α]
[linear_order β]
[topological_space β]
[order_topology β]
[densely_ordered β]
{f : α → β}
(h_mono : monotone f)
(h_dense : dense_range f) :
A monotone function with densely ordered codomain and a dense range is continuous.
theorem
monotone.continuous_of_surjective
{α : Type u}
{β : Type v}
[linear_order α]
[topological_space α]
[order_topology α]
[linear_order β]
[topological_space β]
[order_topology β]
[densely_ordered β]
{f : α → β}
(h_mono : monotone f)
(h_surj : function.surjective f) :
A monotone surjective function with a densely ordered codomain is surjective.
Continuity of order isomorphisms #
In this section we prove that an order_iso is continuous, hence it is a homeomorph. We prove
this for an order_iso between to partial orders with order topology.
@[protected]
theorem
order_iso.continuous
{α : Type u}
{β : Type v}
[partial_order α]
[partial_order β]
[topological_space α]
[topological_space β]
[order_topology α]
[order_topology β]
(e : α ≃o β) :
def
order_iso.to_homeomorph
{α : Type u}
{β : Type v}
[partial_order α]
[partial_order β]
[topological_space α]
[topological_space β]
[order_topology α]
[order_topology β]
(e : α ≃o β) :
α ≃ₜ β
An order isomorphism between two linear order order_topology spaces is a homeomorphism.
Equations
- e.to_homeomorph = {to_equiv := e.to_equiv, continuous_to_fun := _, continuous_inv_fun := _}
@[simp]
theorem
order_iso.coe_to_homeomorph
{α : Type u}
{β : Type v}
[partial_order α]
[partial_order β]
[topological_space α]
[topological_space β]
[order_topology α]
[order_topology β]
(e : α ≃o β) :
⇑(e.to_homeomorph) = ⇑e
@[simp]
theorem
order_iso.coe_to_homeomorph_symm
{α : Type u}
{β : Type v}
[partial_order α]
[partial_order β]
[topological_space α]
[topological_space β]
[order_topology α]
[order_topology β]
(e : α ≃o β) :