Editing Compact space (section)

===Open cover definition===

Formally, a [[topological space]] {{mvar|X}} is called ''compact'' if every [[open cover]] of {{mvar|X}} has a [[finite set|finite]] [[subcover]].<ref>{{cite web |title=Compact Space |last=Weisstein |first=Eric W. |website=Wolfram MathWorld |lang=en |url=http://mathworld.wolfram.com/CompactSpace.html |access-date=2019-11-25}}</ref> That is, {{mvar|X}} is compact if for every collection {{mvar|C}} of open subsets<ref>Here, "collection" means "[[set (mathematics)|set]]" but is used because "collection of open subsets" is less awkward than "set of open subsets". Similarly, "subcollection" means "subset".</ref> of {{mvar|X}} such that

<math display="block">X = \bigcup_{S \in C}S\ ,</math>

there is a '''finite''' subcollection {{mvar|F}} ⊆ {{mvar|C}} such that

<math display="block">X = \bigcup_{S \in F} S\ .</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 space|Hausdorff]] and ''quasi-compact''. A compact set is sometimes referred to as a ''compactum'', plural ''compacta''.