Limit point compact space

Remarks

Every compact space is limit point compact, and so is every countably compact space. For T1 spaces, limit point compactness is equivalent to countable compactness. For metric spaces, it is equivalent to compactness and to sequential compactness. Other names in use are weakly countably compact and Fréchet compact.

Example: $[0, 1]$ is limit point compact. An infinite subset contains a sequence of distinct points, which has a convergent subsequence by the Bolzano–Weierstrass theorem, and its limit is a limit point of the set.

Non-example: $\R$ is not limit point compact. The infinite set $\Z$ has no limit point, since each $x \in \R$ has a neighbourhood $(x - 1/2, x + 1/2)$ that contains at most one integer.

Without the T1 condition, limit point compact need not imply countably compact. Let $X = \N \times \set{0, 1}$, where $\N$ is discrete and $\set{0, 1}$ has the trivial topology. Each nonempty subset has a limit point, since $(n, 1 - i)$ is a limit point of any set that contains $(n, i)$. But the countable open cover by the sets $\set{n} \times \set{0, 1}$ has no finite subcover.