mathlib3 documentation

topology.uniform_space.compact

Compact separated uniform spaces #

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

Main statements #

Implementation notes #

The construction uniform_space_of_compact_t2 is not declared as an instance, as it would badly loop.

tags #

uniform space, uniform continuity, compact space

Uniformity on compact spaces #

On a compact uniform space, the topology determines the uniform structure, entourages are exactly the neighborhoods of the diagonal.

theorem compact_space_uniformity {α : Type u_1} [uniform_space α] [compact_space α] :
uniformity α = (x : α), nhds (x, x)

On a compact uniform space, the topology determines the uniform structure, entourages are exactly the neighborhoods of the diagonal.

The unique uniform structure inducing a given compact topological structure.

Equations

Heine-Cantor theorem #

Heine-Cantor: a continuous function on a compact uniform space is uniformly continuous.

theorem is_compact.uniform_continuous_on_of_continuous {α : Type u_1} {β : Type u_2} [uniform_space α] [uniform_space β] {s : set α} {f : α β} (hs : is_compact s) (hf : continuous_on f s) :

Heine-Cantor: a continuous function on a compact set of a uniform space is uniformly continuous.

theorem is_compact.uniform_continuous_at_of_continuous_at {α : Type u_1} {β : Type u_2} [uniform_space α] [uniform_space β] {r : set × β)} {s : set α} (hs : is_compact s) (f : α β) (hf : (a : α), a s continuous_at f a) (hr : r uniformity β) :
{x : α × α | x.fst s (f x.fst, f x.snd) r} uniformity α

If s is compact and f is continuous at all points of s, then f is "uniformly continuous at the set s", i.e. f x is close to f y whenever x ∈ s and y is close to x (even if y is not itself in s, so this is a stronger assertion than uniform_continuous_on s).

theorem continuous.uniform_continuous_of_tendsto_cocompact {α : Type u_1} {β : Type u_2} [uniform_space α] [uniform_space β] {f : α β} {x : β} (h_cont : continuous f) (hx : filter.tendsto f (filter.cocompact α) (nhds x)) :

If f has compact multiplicative support, then f tends to 1 at infinity.

If f has compact support, then f tends to zero at infinity.

theorem continuous_on.tendsto_uniformly {α : Type u_1} {β : Type u_2} {γ : Type u_3} [uniform_space α] [uniform_space β] [locally_compact_space α] [compact_space β] [uniform_space γ] {f : α β γ} {x : α} {U : set α} (hxU : U nhds x) (h : continuous_on f (U ×ˢ set.univ)) :

A family of functions α → β → γ tends uniformly to its value at x if α is locally compact, β is compact and f is continuous on U × (univ : set β) for some neighborhood U of x.

theorem continuous.tendsto_uniformly {α : Type u_1} {β : Type u_2} {γ : Type u_3} [uniform_space α] [uniform_space β] [locally_compact_space α] [compact_space β] [uniform_space γ] (f : α β γ) (h : continuous f) (x : α) :

A continuous family of functions α → β → γ tends uniformly to its value at x if α is locally compact and β is compact.

An equicontinuous family of functions defined on a compact uniform space is automatically uniformly equicontinuous.