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Compact (Topology)**

A topological space \(X\) is 'compact' if every open cover \((U)_{i\in I}\) of \(X\) admits a finite subcover \((U_{i_n})_{n=1}^N\). A subset \(A\) of \(X\) is a compact subet if \(A\) is compact under the subspace topology. In a Heine-Borel metric space (such as, but not limited to, \(\mathbb{R}^n\)), a subset is compact iff it is closed and bounded.