# mathlibdocumentation

topology.uniform_space.compact_separated

# Compact separated uniform spaces

## Main statements

• compact_space_uniformity: On a separated 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 separated 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 separated spaces

theorem compact_space_uniformity {α : Type u_1}  :
𝓤 α = ⨆ (x : α), 𝓝 (x, x)

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

theorem unique_uniformity_of_compact_t2 {α : Type u_1} [t : topological_space α] [t2_space α] {u u' : uniform_space α}  :
u = u'

def uniform_space_of_compact_t2 {α : Type (max (max u_1 u_2 u_3) u_2)} [t2_space α] :

The unique uniform structure inducing a given compact Hausdorff 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 separated 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) (hs' : is_separated s) (hf : s) :

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