Editing Compact space (section)

== Properties of compact spaces ==

* A compact subset of a [[Hausdorff space]] {{mvar|X}} is closed. 
** If {{mvar|X}} is not Hausdorff then a compact subset of {{mvar|X}} may fail to be a closed subset of {{mvar|X}} (see footnote for example).{{efn|
Let {{math|1=''X'' = {''a'', ''b''}<!---->}} and endow {{mvar|X}} with the topology {{math|{''X'', ∅, {''a''}<!---->}<!---->}}. Then {{math|{''a''}<!---->}} is a compact set but it is not closed.
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** If {{mvar|X}} is not Hausdorff then the closure of a compact set may fail to be compact (see footnote for example).{{efn|
Let {{mvar|X}} be the set of non-negative integers. We endow {{mvar|X}} with the [[particular point topology]] by defining a subset {{math|''U'' ⊆ ''X''}} to be open if and only if {{math|0 ∈ ''U''}}. Then {{math|1=''S'' := {0}<!---->}} is compact, the closure of {{mvar|S}} is all of {{mvar|X}}, but {{mvar|X}} is not compact since the collection of open subsets {{math|{<!---->{0, ''x''} : ''x'' ∈ ''X''}<!---->}} does not have a finite subcover.
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* In any [[topological vector space]] (TVS), a compact subset is [[complete space|complete]]. However, every non-Hausdorff TVS contains compact (and thus complete) subsets that are ''not'' closed.
* If {{mvar|A}} and {{mvar|B}} are disjoint compact subsets of a Hausdorff space {{mvar|X}}, then there exist disjoint open sets {{mvar|U}} and {{mvar|V}} in {{mvar|X}} such that {{math|''A'' ⊆ ''U''}} and {{math|''B'' ⊆ ''V''}}.
* A continuous bijection from a compact space into a Hausdorff space is a [[homeomorphism]].
* A compact Hausdorff space is [[Normal space|normal]] and [[Regular space|regular]].
* If a space {{mvar|X}} is compact and Hausdorff, then no finer topology on {{mvar|X}} is compact and no coarser topology on {{mvar|X}} is Hausdorff.
* If a subset of a metric space {{math|(''X'', ''d'')}} is compact then it is {{mvar|d}}-bounded.

=== Functions and compact spaces ===

Since a [[continuous function (topology)|continuous]] image of a compact space is compact, the [[extreme value theorem]] holds for such spaces: a continuous real-valued function on a nonempty compact space is bounded above and attains its supremum.<ref>{{harvnb|Arkhangel'skii|Fedorchuk|1990|loc=Corollary 5.2.1}}</ref> 
(Slightly more generally, this is true for an upper semicontinuous function.) As a sort of converse to the above statements, the pre-image of a compact space under a [[proper map]] is compact.

=== Compactifications ===
{{Main|Compactification (mathematics)}}

Every topological space {{mvar|X}} is an open [[dense topological subspace|dense subspace]] of a compact space having at most one point more than {{mvar|X}}, by the [[Alexandroff one-point compactification]]. 
By the same construction, every [[locally compact]] Hausdorff space {{mvar|X}} is an open dense subspace of a compact Hausdorff space having at most one point more than {{mvar|X}}.

=== Ordered compact spaces ===

A nonempty compact subset of the [[real number]]s has a greatest element and a least element.

Let {{mvar|X}} be a [[total order|simply ordered]] set endowed with the [[order topology]]. 
Then {{mvar|X}} is compact if and only if {{mvar|X}} is a [[complete lattice]] (i.e. all subsets have suprema and infima).<ref>{{harvnb|Steen|Seebach|1995|p=67}}</ref>