📂MetricSpace

Equivalence of Various Compactnesses in Metric Spaces

Theorem¹

Let $X$ be a metric space. The following propositions are equivalent.

• $X$ is compact.
• $X$ is countably compact.
• $X$ is limit point compact.
• $X$ is sequentially compact.

Explanation

This generally does not hold in topological spaces but holds in metric spaces.