Alternate Definition: A subspace A of X is compact if and only if every open cover of A by open sets in X has a finite subcover.

How do you prove a subspace is compact?

Let Y be a subspace of X. Then Y is compact if and only if every covering of Y by sets open in X contains a finite sub-collection covering Y . Proof. Suppose that Y is compact and A = {Aα}α∈J is a covering of Y by sets open in X.

What is a compact subspace?

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 , there is a finite subset F of C such that . Compactness is a "topological" property.