# Compactness and Open Covers

Let \(S \subseteq \mathbf{R}^n\) and suppose \(\mathcal{A} = {A_i | i \in I}\) is a family of open subsets of \(\mathbf{R}^n\) such that \[ S \subseteq \bigcup_{i \in I} A_i. \] We call such a family \(\mathcal{A}\) an open cover of \(S\). We call a subset \(K \subseteq \mathbf{R}^n\) topologically compact if every open cover \(\mathcal{A}\) of \(K\) admits a finite subcover. That is for every open cover \(\mathcal{A}\), there exists \(k \in \mathbf{N}\) such that there exist \(A_1,…