Editing Compact space (section)

==== Metric spaces ====

For any metric space {{math|(''X'', ''d'')}}, the following are equivalent (assuming [[countable choice]]):
# {{math|(''X'', ''d'')}} is compact.
# {{math|(''X'', ''d'')}} is [[completeness (topology)|complete]] and [[totally bounded]] (this is also equivalent to compactness for [[uniform space]]s).<ref>{{harvnb|Arkhangel'skii|Fedorchuk|1990|loc=Theorem&nbsp;5.3.7}}</ref>
# {{math|(''X'', ''d'')}} is sequentially compact; that is, every [[sequence]] in {{mvar|X}} has a convergent subsequence whose limit is in {{mvar|X}} (this is also equivalent to compactness for [[first-countable]] [[uniform space]]s).
# {{math|(''X'', ''d'')}} is [[limit point compact]] (also called weakly countably compact); that is, every infinite subset of {{mvar|X}} has at least one [[Limit point of a set|limit point]] in {{mvar|X}}.
# {{math|(''X'', ''d'')}} is [[countably compact]]; that is, every countable open cover of {{mvar|X}} has a finite subcover.
# {{math|(''X'', ''d'')}} is an image of a continuous function from the [[Cantor set]].<ref>{{harvnb|Willard|1970}} Theorem&nbsp;30.7.</ref>
# Every decreasing nested sequence of nonempty closed subsets {{math|''S''<sub>1</sub> ⊇ ''S''<sub>2</sub> ⊇ ...}} in {{math|(''X'', ''d'')}} has a nonempty intersection.
# Every increasing nested sequence of proper open subsets {{math|''S''<sub>1</sub> ⊆ ''S''<sub>2</sub> ⊆ ...}} in {{math|(''X'', ''d'')}} fails to cover {{mvar|X}}.

A compact metric space {{math|(''X'', ''d'')}} also satisfies the following properties:
# [[Lebesgue's number lemma]]: For every open cover of {{mvar|X}}, there exists a number {{nowrap|''δ'' > 0}} such that every subset of {{mvar|X}} of diameter < {{mvar|δ}} is contained in some member of the cover.
# {{math|(''X'', ''d'')}} is [[second-countable space|second-countable]], [[Separable space|separable]] and [[Lindelöf space|Lindelöf]] – these three conditions are equivalent for metric spaces. The converse is not true; e.g., a countable discrete space satisfies these three conditions, but is not compact.
# {{mvar|X}} is closed and bounded (as a subset of any metric space whose restricted metric is {{mvar|d}}). The converse may fail for a non-Euclidean space; e.g. the [[real line]] equipped with the [[discrete metric]] is closed and bounded but not compact, as the collection of all [[Singleton (mathematics)|singletons]] of the space is an open cover which admits no finite subcover. It is complete but not totally bounded.