16 - Equivalence Statements for Compactness

Equivalence Statements for Compactness

Let $X$ be a metric space, then the following are equivalent

1. $X$ is compact

2. $X$ is sequentially compact

3. $X$ is complete and totally bounded

Proof

As a consequence of

1. Compactness implies sequentially compactness
2. Sequentially Compactness Characterisation Theorem