# mathlib3documentation

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 #

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

• uniform_space_of_compact_t2: every compact T2 topological structure is induced by a uniform structure. This uniform structure is described in the previous item.

• Heine-Cantor theorem: continuous functions on compact uniform spaces with values in uniform spaces are automatically uniformly continuous. There are several variations, the main one is compact_space.uniform_continuous_of_continuous.

## 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 #

theorem nhds_set_diagonal_eq_uniformity {α : Type u_1}  :

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}  :
= (x : α), nhds (x, x)

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

theorem unique_uniformity_of_compact {γ : Type u_3} [t : topological_space γ] {u u' : uniform_space γ}  :
u = u'
def uniform_space_of_compact_t2 {γ : Type u_3} [t2_space γ] :

The unique uniform structure inducing a given compact topological structure.

Equations

### Heine-Cantor theorem #

theorem compact_space.uniform_continuous_of_continuous {α : Type u_1} {β : Type u_2} {f : α β} (h : continuous f) :

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} {s : set α} {f : α β} (hs : is_compact s) (hf : 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} {r : set × β)} {s : set α} (hs : is_compact s) (f : α β) (hf : (a : α), a s ) (hr : r ) :
{x : α × α | x.fst s (f x.fst, f x.snd) r}

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} {f : α β} {x : β} (h_cont : continuous f) (hx : (nhds x)) :
theorem has_compact_mul_support.is_one_at_infty {α : Type u_1} {γ : Type u_3} {f : α γ} [has_one γ] (h : has_compact_mul_support f) :
(nhds 1)

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

theorem has_compact_support.is_zero_at_infty {α : Type u_1} {γ : Type u_3} {f : α γ} [has_zero γ] (h : has_compact_support f) :
(nhds 0)

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

theorem has_compact_support.uniform_continuous_of_continuous {α : Type u_1} {β : Type u_2} {f : α β} [has_zero β] (h1 : has_compact_support f) (h2 : continuous f) :
theorem has_compact_mul_support.uniform_continuous_of_continuous {α : Type u_1} {β : Type u_2} {f : α β} [has_one β] (h1 : has_compact_mul_support f) (h2 : continuous f) :
theorem continuous_on.tendsto_uniformly {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α β γ} {x : α} {U : set α} (hxU : U nhds x) (h : (U ×ˢ set.univ)) :
(f x) (nhds x)

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} (f : α β γ) (h : continuous f) (x : α) :
(f x) (nhds x)

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

theorem compact_space.uniform_equicontinuous_of_equicontinuous {α : Type u_1} {β : Type u_2} {ι : Type u_3} {F : ι β α} (h : equicontinuous F) :

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