File: Mathlib/Topology/Algebra/WithZeroTopology.lean

full_name: WithZeroTopology.topologicalSpace
code: /-- 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. -/
scoped instance (priority := 100) topologicalSpace : TopologicalSpace Γ₀ 
goal: TopologicalSpace Γ₀

full_name: WithZeroTopology.nhds_eq_update
code: theorem nhds_eq_update : (𝓝 : Γ₀ → Filter Γ₀) = update pure 0 (⨅ γ ≠ 0, 𝓟 (Iio γ)) 
goal: 𝓝 = Function.update pure 0 (⨅ γ, ⨅ (_ : γ ≠ 0), 𝓟 (Set.Iio γ))

full_name: WithZeroTopology.nhds_zero
code: theorem nhds_zero : 𝓝 (0 : Γ₀) = ⨅ γ ≠ 0, 𝓟 (Iio γ) 
goal: 𝓝 0 = ⨅ γ, ⨅ (_ : γ ≠ 0), 𝓟 (Set.Iio γ)

full_name: WithZeroTopology.hasBasis_nhds_zero
code: /-- In a linearly ordered group with zero element adjoined, `U` is a neighbourhood of `0` if and
only if there exists a nonzero element `γ₀` such that `Iio γ₀ ⊆ U`. -/
theorem hasBasis_nhds_zero : (𝓝 (0 : Γ₀)).HasBasis (fun γ : Γ₀ => γ ≠ 0) Iio 
goal: (𝓝 0).HasBasis (fun γ => γ ≠ 0) Set.Iio

full_name: WithZeroTopology.Iio_mem_nhds_zero
code: theorem Iio_mem_nhds_zero (hγ : γ ≠ 0) : Iio γ ∈ 𝓝 (0 : Γ₀) 
goal: Set.Iio γ ∈ 𝓝 0

full_name: WithZeroTopology.nhds_zero_of_units
code: /-- If `γ` is an invertible element of a linearly ordered group with zero element adjoined, then
`Iio (γ : Γ₀)` is a neighbourhood of `0`. -/
theorem nhds_zero_of_units (γ : Γ₀ˣ) : Iio ↑γ ∈ 𝓝 (0 : Γ₀) 
goal: Set.Iio ↑γ ∈ 𝓝 0

full_name: WithZeroTopology.tendsto_zero
code: theorem tendsto_zero : Tendsto f l (𝓝 (0 : Γ₀)) ↔ ∀ (γ₀) (_ : γ₀ ≠ 0), ∀ᶠ x in l, f x < γ₀ 
goal: Filter.Tendsto f l (𝓝 0) ↔ ∀ (γ₀ : Γ₀), γ₀ ≠ 0 → ∀ᶠ (x : α) in l, f x < γ₀

full_name: WithZeroTopology.nhds_of_ne_zero
code: /-- The neighbourhood filter of a nonzero element consists of all sets containing that
element. -/
@[simp]
theorem nhds_of_ne_zero {γ : Γ₀} (h₀ : γ ≠ 0) : 𝓝 γ = pure γ 
goal: 𝓝 γ = pure γ

full_name: WithZeroTopology.nhds_coe_units
code: /-- The neighbourhood filter of an invertible element consists of all sets containing that
element. -/
theorem nhds_coe_units (γ : Γ₀ˣ) : 𝓝 (γ : Γ₀) = pure (γ : Γ₀) 
goal: 𝓝 ↑γ = pure ↑γ

full_name: WithZeroTopology.singleton_mem_nhds_of_units
code: /-- If `γ` is an invertible element of a linearly ordered group with zero element adjoined, then
`{γ}` is a neighbourhood of `γ`. -/
theorem singleton_mem_nhds_of_units (γ : Γ₀ˣ) : ({↑γ} : Set Γ₀) ∈ 𝓝 (γ : Γ₀) 
goal: {↑γ} ∈ 𝓝 ↑γ

full_name: WithZeroTopology.singleton_mem_nhds_of_ne_zero
code: /-- If `γ` is a nonzero element of a linearly ordered group with zero element adjoined, then `{γ}`
is a neighbourhood of `γ`. -/
theorem singleton_mem_nhds_of_ne_zero (h : γ ≠ 0) : ({γ} : Set Γ₀) ∈ 𝓝 (γ : Γ₀) 
goal: {γ} ∈ 𝓝 γ

full_name: WithZeroTopology.hasBasis_nhds_of_ne_zero
code: theorem hasBasis_nhds_of_ne_zero {x : Γ₀} (h : x ≠ 0) :
    HasBasis (𝓝 x) (fun _ : Unit => True) fun _ => {x} 
goal: (𝓝 x).HasBasis (fun x => True) fun x_1 => {x}

full_name: WithZeroTopology.hasBasis_nhds_units
code: theorem hasBasis_nhds_units (γ : Γ₀ˣ) :
    HasBasis (𝓝 (γ : Γ₀)) (fun _ : Unit => True) fun _ => {↑γ} 
goal: (𝓝 ↑γ).HasBasis (fun x => True) fun x => {↑γ}

full_name: WithZeroTopology.tendsto_of_ne_zero
code: theorem tendsto_of_ne_zero {γ : Γ₀} (h : γ ≠ 0) : Tendsto f l (𝓝 γ) ↔ ∀ᶠ x in l, f x = γ 
goal: Filter.Tendsto f l (𝓝 γ) ↔ ∀ᶠ (x : α) in l, f x = γ

full_name: WithZeroTopology.tendsto_units
code: theorem tendsto_units {γ₀ : Γ₀ˣ} : Tendsto f l (𝓝 (γ₀ : Γ₀)) ↔ ∀ᶠ x in l, f x = γ₀ 
goal: Filter.Tendsto f l (𝓝 ↑γ₀) ↔ ∀ᶠ (x : α) in l, f x = ↑γ₀

full_name: WithZeroTopology.Iio_mem_nhds
code: theorem Iio_mem_nhds (h : γ₁ < γ₂) : Iio γ₂ ∈ 𝓝 γ₁ 
goal: Set.Iio γ₂ ∈ 𝓝 γ₁

full_name: WithZeroTopology.isOpen_iff
code: theorem isOpen_iff {s : Set Γ₀} : IsOpen s ↔ (0 : Γ₀) ∉ s ∨ ∃ γ, γ ≠ 0 ∧ Iio γ ⊆ s 
goal: IsOpen s ↔ 0 ∉ s ∨ ∃ γ, γ ≠ 0 ∧ Set.Iio γ ⊆ s

full_name: WithZeroTopology.isClosed_iff
code: theorem isClosed_iff {s : Set Γ₀} : IsClosed s ↔ (0 : Γ₀) ∈ s ∨ ∃ γ, γ ≠ 0 ∧ s ⊆ Ici γ 
goal: IsClosed s ↔ 0 ∈ s ∨ ∃ γ, γ ≠ 0 ∧ s ⊆ Set.Ici γ

full_name: WithZeroTopology.isOpen_Iio
code: theorem isOpen_Iio {a : Γ₀} : IsOpen (Iio a) 
goal: IsOpen (Set.Iio a)

full_name: WithZeroTopology.orderClosedTopology
code: /-- The topology on a linearly ordered group with zero element adjoined is compatible with the order
structure: the set `{p : Γ₀ × Γ₀ | p.1 ≤ p.2}` is closed. -/
@[nolint defLemma]
scoped instance (priority := 100) orderClosedTopology : OrderClosedTopology Γ₀ 
goal: OrderClosedTopology Γ₀

full_name: WithZeroTopology.t5Space
code: /-- The topology on a linearly ordered group with zero element adjoined is T₅. -/
@[nolint defLemma]
scoped instance (priority := 100) t5Space : T5Space Γ₀ 
goal: T5Space Γ₀

full_name: WithZeroTopology.instContinuousMul
code: /-- The topology on a linearly ordered group with zero element adjoined makes it a topological
monoid. -/
@[nolint defLemma]
scoped instance (priority := 100) : ContinuousMul Γ₀ 
goal: ContinuousMul Γ₀

full_name: WithZeroTopology.instHasContinuousInv₀
code: @[nolint defLemma]
scoped instance (priority := 100) : HasContinuousInv₀ Γ₀ 
goal: HasContinuousInv₀ Γ₀