Editing Compact space (section)

=== Compactness of subsets ===

A subset {{mvar|K}} of a topological space {{mvar|X}} is said to be compact if it is compact as a subspace (in the [[subspace topology]]). That is, {{mvar|K}} is compact if for every arbitrary collection {{mvar|C}} of open subsets of {{mvar|X}} such that

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

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

<math display="block">K \subseteq \bigcup_{S \in F} S\ .</math>

Because compactness is a [[topological property]], the compactness of a subset depends only on the subspace topology induced on it.  It follows that, if <math>K \subset Z \subset Y</math>, with subset {{mvar|Z}} equipped with the subspace topology, then {{mvar|K}} is compact in {{mvar|Z}} if and only if {{mvar|K}} is compact in {{mvar|Y}}.