# mathlib3documentation

topology.algebra.field

# Topological fields #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

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 (λ (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 (λ (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]
structure topological_division_ring (K : Type u_1)  :
Prop
• to_topological_ring :
• to_has_continuous_inv₀ :

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

Instances of this typeclass
def subfield.topological_closure {α : Type u_2} [field α] (K : subfield α) :

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

Equations
theorem subfield.le_topological_closure {α : Type u_2} [field α] (s : subfield α) :
theorem subfield.is_closed_topological_closure {α : Type u_2} [field α] (s : subfield α) :
theorem subfield.topological_closure_minimal {α : Type u_2} [field α] (s : subfield α) {t : subfield α} (h : s t) (ht : is_closed 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.

def affine_homeomorph {𝕜 : Type u_2} [field 𝕜] (a b : 𝕜) (h : a 0) :
𝕜 ≃ₜ 𝕜

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

Equations
@[simp]
theorem affine_homeomorph_apply {𝕜 : Type u_2} [field 𝕜] (a b : 𝕜) (h : a 0) (x : 𝕜) :
h) x = a * x + b
@[simp]
theorem affine_homeomorph_symm_apply {𝕜 : Type u_2} [field 𝕜] (a b : 𝕜) (h : a 0) (y : 𝕜) :
b h).symm) y = (y - b) / a
theorem is_local_min.inv {α : Type u_2} {β : Type u_3} {f : α β} {a : α} (h1 : a) (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 is_preconnected.eq_one_or_eq_neg_one_of_sq_eq {α : Type u_2} {𝕜 : Type u_3} {f : α 𝕜} {S : set α} [t1_space 𝕜] [ring 𝕜] (hS : is_preconnected S) (hf : S) (hsq : set.eq_on (f ^ 2) 1 S) :
1 S (-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 is_preconnected.eq_or_eq_neg_of_sq_eq {α : Type u_2} {𝕜 : Type u_3} {f g : α 𝕜} {S : set α} [t1_space 𝕜] [field 𝕜] (hS : is_preconnected S) (hf : S) (hg : S) (hsq : set.eq_on (f ^ 2) (g ^ 2) S) (hg_ne : {x : α}, x S g x 0) :
g S (-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 is_preconnected.eq_of_sq_eq {α : Type u_2} {𝕜 : Type u_3} {f g : α 𝕜} {S : set α} [t1_space 𝕜] [field 𝕜] (hS : is_preconnected S) (hf : S) (hg : S) (hsq : set.eq_on (f ^ 2) (g ^ 2) S) (hg_ne : {x : α}, x S g x 0) {y : α} (hy : y S) (hy' : f y = g y) :
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.