Editing General topology (section)

==Compact sets==
{{Main|Compact (mathematics)}}
Formally, a [[topological space]] ''X'' is called ''compact'' if each of its [[open cover]]s has a [[finite set|finite]] [[subcover]]. Otherwise it is called ''non-compact''.  Explicitly, this means that for every arbitrary collection

:<math>\{U_\alpha\}_{\alpha\in A}</math>

of open subsets of {{mvar|X}} such that

:<math>X = \bigcup_{\alpha\in A} U_\alpha,</math>

there is a finite subset {{mvar|J}} of {{mvar|A}} such that

:<math>X = \bigcup_{i\in J} U_i.</math>

Some branches of mathematics such as [[algebraic geometry]], typically influenced by the French school of [[Nicolas Bourbaki|Bourbaki]], use the term ''quasi-compact'' for the general notion, and reserve the term ''compact'' for topological spaces that are both [[Hausdorff spaces|Hausdorff]] and ''quasi-compact''.  A compact set is sometimes referred to as a ''compactum'', plural ''compacta''.

Every closed [[interval (mathematics)|interval]] in '''[[Real number|R]]''' of finite length is [[compact space|compact]]. More is true: In '''R'''<sup><var>n</var></sup>, a set is compact [[if and only if]] it is [[closed set|closed]] and bounded. (See [[Heine–Borel theorem]]).

Every continuous image of a compact space is compact.

A compact subset of a Hausdorff space is closed.

Every continuous [[bijection]] from a compact space to a Hausdorff space is necessarily a [[homeomorphism]].

Every [[sequence]] of points in a compact metric space has a convergent subsequence.

Every compact finite-dimensional [[manifold]] can be embedded in some Euclidean space '''R'''<sup><var>n</var></sup>.