Sequentially compact set

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).