Theorem
101
[0V3] Given a metric space \((X,d)\) and its subset \(Cβ X\), The following three conditions are equivalent.
\(C\) is sequentially compact: every sequence \((x_ n)β C\) has a subsequence converging to an element of \(C\).
\(C\) is compact: from each family of open sets whose union covers \(C\), we can choose a finite subfamily whose union covers \(C\).
\(C\) is complete, and is totally bounded: for every \(π{\gt}0\) there are finite points \(x_ 1...x_ nβ C\) such that \(Cβ β_{i=1}^ n B(x_ i,π)\).