# mathlibdocumentation

topology.algebra.with_zero_topology

# The topology on linearly ordered commutative groups with zero #

Let Γ₀ be a linearly ordered commutative group to which we have adjoined a zero element. Then Γ₀ may naturally be endowed with a topology that turns Γ₀ into a topological monoid. Neighborhoods of zero are sets containing {γ | γ < γ₀} for some invertible element γ₀ and every invertible element is open. In particular the topology is the following: "a subset U ⊆ Γ₀ is open if 0 ∉ U or if there is an invertible γ₀ ∈ Γ₀ such that {γ | γ < γ₀} ⊆ U", but this fact is not proven here since the neighborhoods description is what is actually useful.

We prove this topology is ordered and T₃ (in addition to be compatible with the monoid structure).

All this is useful to extend a valuation to a completion. This is an abstract version of how the absolute value (resp. p-adic absolute value) on ℚ is extended to ℝ (resp. ℚₚ).

## Implementation notes #

This topology is not defined as an instance since it may not be the desired topology on a linearly ordered commutative group with zero. You can locally activate this topology using local attribute [instance] linear_ordered_comm_group_with_zero.topological_space All other instances will (ordered_topology, t3_space, has_continuous_mul) then follow.

def linear_ordered_comm_group_with_zero.nhds_fun (Γ₀ : Type u_1) (x : Γ₀) :
filter Γ₀

The neighbourhoods around γ ∈ Γ₀, used in the definition of the topology on Γ₀. These neighbourhoods are defined as follows: A set s is a neighbourhood of 0 if there is an invertible γ₀ ∈ Γ₀ such that {γ | γ < γ₀} ⊆ s. If γ ≠ 0, then every set that contains γ is a neighbourhood of γ.

Equations
@[protected]

The topology on a linearly ordered commutative group with a zero element adjoined. A subset U is open if 0 ∉ U or if there is an invertible element γ₀ such that {γ | γ < γ₀} ⊆ U.

Equations
theorem linear_ordered_comm_group_with_zero.directed_lt (Γ₀ : Type u_1)  :
(λ (γ₀ : Γ₀ˣ), filter.principal {γ : Γ₀ | γ < γ₀})

The neighbourhoods {γ | γ < γ₀} of 0 form a directed set indexed by the invertible elements γ₀.

At all points of a linearly ordered commutative group with a zero element adjoined, the pure filter is smaller than the filter given by nhds_fun.

theorem linear_ordered_comm_group_with_zero.nhds_fun_ok (Γ₀ : Type u_1) (x : Γ₀) {s : set Γ₀} (s_in : s ) :
∃ (t : set Γ₀) (H : , t s ∀ (y : Γ₀), y t

For every point Γ₀, and every “neighbourhood” s of it (described by nhds_fun), there is a smaller “neighbourhood” t ⊆ s, such that s is a “neighbourhood“ of all the points in t.

theorem linear_ordered_comm_group_with_zero.nhds_coe_units {Γ₀ : Type u_1} (γ : Γ₀ˣ) :

The neighbourhood filter of an invertible element consists of all sets containing that element.

@[simp]
theorem linear_ordered_comm_group_with_zero.nhds_of_ne_zero {Γ₀ : Type u_1} (γ : Γ₀) (h : γ 0) :

The neighbourhood filter of a nonzero element consists of all sets containing that element.

theorem linear_ordered_comm_group_with_zero.singleton_nhds_of_units {Γ₀ : Type u_1} (γ : Γ₀ˣ) :
{γ}

If γ is an invertible element of a linearly ordered group with zero element adjoined, then {γ} is a neighbourhood of γ.

theorem linear_ordered_comm_group_with_zero.singleton_nhds_of_ne_zero {Γ₀ : Type u_1} (γ : Γ₀) (h : γ 0) :
{γ} nhds γ

If γ is a nonzero element of a linearly ordered group with zero element adjoined, then {γ} is a neighbourhood of γ.

theorem linear_ordered_comm_group_with_zero.has_basis_nhds_zero {Γ₀ : Type u_1}  :
(nhds 0).has_basis (λ (_x : Γ₀ˣ), true) (λ (γ₀ : Γ₀ˣ), {γ : Γ₀ | γ < γ₀})

If U is a neighbourhood of 0 in a linearly ordered group with zero element adjoined, then there exists an invertible element γ₀ such that {γ | γ < γ₀} ⊆ U.

theorem linear_ordered_comm_group_with_zero.nhds_zero_of_units {Γ₀ : Type u_1} (γ : Γ₀ˣ) :
{x : Γ₀ | x < γ} nhds 0

If γ is an invertible element of a linearly ordered group with zero element adjoined, then {x | x < γ} is a neighbourhood of 0.

theorem linear_ordered_comm_group_with_zero.tendsto_zero {Γ₀ : Type u_1} {α : Type u_2} {F : filter α} {f : α → Γ₀} :
(nhds 0) ∀ (γ₀ : Γ₀ˣ), {x : α | f x < γ₀} F
theorem linear_ordered_comm_group_with_zero.nhds_zero_of_ne_zero {Γ₀ : Type u_1} (γ : Γ₀) (h : γ 0) :
{x : Γ₀ | x < γ} nhds 0

If γ is a nonzero element of a linearly ordered group with zero element adjoined, then {x | x < γ} is a neighbourhood of 0.

theorem linear_ordered_comm_group_with_zero.has_basis_nhds_units {Γ₀ : Type u_1} (γ : Γ₀ˣ) :
(nhds γ).has_basis (λ (i : unit), true) (λ (i : unit), {γ})
theorem linear_ordered_comm_group_with_zero.has_basis_nhds_of_ne_zero {Γ₀ : Type u_1} {x : Γ₀} (h : x 0) :
(nhds x).has_basis (λ (i : unit), true) (λ (i : unit), {x})
theorem linear_ordered_comm_group_with_zero.singleton_mem_nhds_of_ne_zero {Γ₀ : Type u_1} {x : Γ₀} (h : x 0) :
{x} nhds x
theorem linear_ordered_comm_group_with_zero.tendsto_units {Γ₀ : Type u_1} {α : Type u_2} {F : filter α} {f : α → Γ₀} {γ₀ : Γ₀ˣ} :
(nhds γ₀) {x : α | f x = γ₀} F
theorem linear_ordered_comm_group_with_zero.tendsto_of_ne_zero {Γ₀ : Type u_1} {α : Type u_2} {F : filter α} {f : α → Γ₀} {γ : Γ₀} (h : γ 0) :
(nhds γ) {x : α | f x = γ} F
@[protected, instance]

The topology on a linearly ordered group with zero element adjoined is compatible with the order structure.

@[protected, instance]
def linear_ordered_comm_group_with_zero.t3_space (Γ₀ : Type u_1)  :
t3_space Γ₀

The topology on a linearly ordered group with zero element adjoined is T₃.

@[protected, instance]

The topology on a linearly ordered group with zero element adjoined makes it a topological monoid.