Throughout, \(X\) is a metric space. State the definition of what it means for a set \(K\subset X\) to be compact, including the definition of open cover.
Solution
1. Definitions.
1.1. An open cover of \(K \subseteq X\) is a collection \(\{U_\alpha\}_{\alpha \in A}\) of open subsets of \(X\) such that \(K \subseteq \bigcup_{\alpha \in A} U_\alpha\).
1.2. A finite subcover of an open cover \(\{U_\alpha\}\) is a finite subcollection \(\{U_{\alpha_1}, \ldots, U_{\alpha_m}\}\) with \(K \subseteq \bigcup_{j=1}^m U_{\alpha_j}\).
1.3. \(K\) is compact if every open cover of \(K\) has a finite subcover.
2. Q.E.D.