Sequentially compact

Definition · applies to topological spaces

Graduate (topological spaces)

A topological space $T$ is sequentially compact if every sequence in $T$ has a convergent subsequence.

Examples

$[0, 1]$ High school
$[0, 1]$ is sequentially compact (as a compact metric space), locally compact, and paracompact, because it is compact.

Things that may be mistaken for examples.

The set of rational numbers, $\mathbb{Q}$ High school
The set of rational numbers, $\mathbb{Q}$, is not locally compact, because no neighborhood of a point has a compact closure, so it is not compact or sequentially compact either.
The set of real numbers, $\reals$ High school
The set of real numbers, $\reals$, is not sequentially compact, because the sequence $1, 2, 3, \dots$ has no convergent subsequence.
The closed unit ball of $\ell^2$ Graduate
The closed unit ball of $\ell^2$ is not sequentially compact, because the standard basis vectors have no convergent subsequence.

For metric spaces, sequential compactness is equivalent to compactness. For general topological spaces, neither implies the other.