There are multiple definition on whether a metric space is compact.

Sequentially Compact

A metric space is sequential compact iff every sequence in has at least one convergent subsequence. A subset of a metric space is sequential compact if the subspace is sequentially compact.

With sequentially compact set, we can get Heine-Borel theorem to work under metric space.

Topological Compact

A metric space is compact if every open cover of has a finite subcover.

Continuous Maps Preserve Compactness

Let be a continuous maps from one metric space to . Let be any compact subset . Then the image of is also compact.