Sequentially compact set - Imported Knowlpedia Content

A sequentially compact set is a subset $K\subseteq X$ of a topological space $X$ such that every sequence $(x_n)$ in $K$ has a subsequence $(x_{n_k})$ that is a convergent sequence in $X$ with limit $x\in K$ .

Sequential compactness is phrased purely in terms of sequences, and in many important settings (notably metric spaces) it closely tracks compactness.

Examples

In $\mathbb{R}$ with the usual topology, $[0,1]$ is sequentially compact.
An infinite set with the discrete topology is not sequentially compact (a sequence of distinct points has no convergent subsequence).