Is compactness a generalization of completeness

Question

Is the concept of compact spaces a generalization of completeness to non-metric topological spaces?

Answer 1

Not quite. A metric space space can be complete without being compact (e.g., $\mathbb R$ with the Euclidean topology). For a metric space, completeness + total boundedness = compactness.

Answer 2

Generalization of completeness to non-metric spaces goes through the concept of uniform spaces.