Limit Point Compactness

Subjects: Topology
Links: Countable compactness, Limit Points and Closure, Compactness-Type Properties

Def: A space $X$ is said to be limit point compact or weakly countably compact if every infinite subset of $X$ has a limit point.

Prop: Countable compactness implies limit point compactness.

Cor: Compactness implies limit point compactness.

Lemma: For first countable Hausdorff spaces, limit point compactness implies Sequential Compactness

Th: For metric spaces and second countable Hausdorff spaces, limit point compactness, sequential compactness, and compactness are all equivalent.