# Topological fields #

A topological division ring is a topological ring whose inversion function is continuous at every non-zero element.

theorem Filter.tendsto_cocompact_mul_left₀ {K : Type u_1} [] [] [] {a : K} (ha : a 0) :
Filter.Tendsto (fun (x : K) => a * x) () ()

Left-multiplication by a nonzero element of a topological division ring is proper, i.e., inverse images of compact sets are compact.

theorem Filter.tendsto_cocompact_mul_right₀ {K : Type u_1} [] [] [] {a : K} (ha : a 0) :
Filter.Tendsto (fun (x : K) => x * a) () ()

Right-multiplication by a nonzero element of a topological division ring is proper, i.e., inverse images of compact sets are compact.

class TopologicalDivisionRing (K : Type u_1) [] [] extends , :

A topological division ring is a division ring with a topology where all operations are continuous, including inversion.

Instances
def Subfield.topologicalClosure {α : Type u_2} [] [] (K : ) :

The (topological-space) closure of a subfield of a topological field is itself a subfield.

Equations
• K.topologicalClosure = let __src := K.topologicalClosure; { carrier := closure K, mul_mem' := , one_mem' := , add_mem' := , zero_mem' := , neg_mem' := , inv_mem' := }
Instances For
theorem Subfield.le_topologicalClosure {α : Type u_2} [] [] (s : ) :
s s.topologicalClosure
theorem Subfield.isClosed_topologicalClosure {α : Type u_2} [] [] (s : ) :
IsClosed s.topologicalClosure
theorem Subfield.topologicalClosure_minimal {α : Type u_2} [] [] (s : ) {t : } (h : s t) (ht : IsClosed t) :
s.topologicalClosure t

This section is about affine homeomorphisms from a topological field 𝕜 to itself. Technically it does not require 𝕜 to be a topological field, a topological ring that happens to be a field is enough.

@[simp]
theorem affineHomeomorph_apply {𝕜 : Type u_2} [] [] [] (a : 𝕜) (b : 𝕜) (h : a 0) (x : 𝕜) :
() x = a * x + b
@[simp]
theorem affineHomeomorph_symm_apply {𝕜 : Type u_2} [] [] [] (a : 𝕜) (b : 𝕜) (h : a 0) (y : 𝕜) :
().symm y = (y - b) / a
def affineHomeomorph {𝕜 : Type u_2} [] [] [] (a : 𝕜) (b : 𝕜) (h : a 0) :
𝕜 ≃ₜ 𝕜

The map fun x => a * x + b, as a homeomorphism from 𝕜 (a topological field) to itself, when a ≠ 0.

Equations
• = { toFun := fun (x : 𝕜) => a * x + b, invFun := fun (y : 𝕜) => (y - b) / a, left_inv := , right_inv := , continuous_toFun := , continuous_invFun := }
Instances For
theorem IsLocalMin.inv {α : Type u_2} {β : Type u_3} [] {f : αβ} {a : α} (h1 : ) (h2 : ∀ᶠ (z : α) in nhds a, 0 < f z) :

Some results about functions on preconnected sets valued in a ring or field with a topology.

theorem IsPreconnected.eq_one_or_eq_neg_one_of_sq_eq {α : Type u_2} {𝕜 : Type u_3} {f : α𝕜} {S : Set α} [] [] [] [Ring 𝕜] [] (hS : ) (hf : ) (hsq : Set.EqOn (f ^ 2) 1 S) :
Set.EqOn f 1 S Set.EqOn f (-1) S

If f is a function α → 𝕜 which is continuous on a preconnected set S, and f ^ 2 = 1 on S, then either f = 1 on S, or f = -1 on S.

theorem IsPreconnected.eq_or_eq_neg_of_sq_eq {α : Type u_2} {𝕜 : Type u_3} {f : α𝕜} {g : α𝕜} {S : Set α} [] [] [] [] [] (hS : ) (hf : ) (hg : ) (hsq : Set.EqOn (f ^ 2) (g ^ 2) S) (hg_ne : ∀ {x : α}, x Sg x 0) :
Set.EqOn f g S Set.EqOn f (-g) S

If f, g are functions α → 𝕜, both continuous on a preconnected set S, with f ^ 2 = g ^ 2 on S, and g z ≠ 0 all z ∈ S, then either f = g or f = -g on S.

theorem IsPreconnected.eq_of_sq_eq {α : Type u_2} {𝕜 : Type u_3} {f : α𝕜} {g : α𝕜} {S : Set α} [] [] [] [] [] (hS : ) (hf : ) (hg : ) (hsq : Set.EqOn (f ^ 2) (g ^ 2) S) (hg_ne : ∀ {x : α}, x Sg x 0) {y : α} (hy : y S) (hy' : f y = g y) :
Set.EqOn f g S

If f, g are functions α → 𝕜, both continuous on a preconnected set S, with f ^ 2 = g ^ 2 on S, and g z ≠ 0 all z ∈ S, then as soon as f = g holds at one point of S it holds for all points.