Compact space
Compactness of subsets

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

K ⊆ ⋃ S ∈ C S , {\displaystyle K\subseteq \bigcup _{S\in C}S\,}

there is a finite subcollection F ⊆ C such that

K ⊆ ⋃ S ∈ F S. {\displaystyle K\subseteq \bigcup _{S\in F}S\.}

Compactness is a topological property. That is, if K ⊂ Z ⊂ Y {\displaystyle K\subset Z\subset Y}, with subset Z equipped with the subspace topology, then K is compact in Z if and only if K is compact in Y.
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