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.
박대희·안승호, 위상수학(하) (5/E, 2022), p503 ↩︎