📂Topology

What is Sequential Compactness in Topological Spaces?

Definition¹

A topological space $X$ is said to be sequentially compact if every sequence $\left\{ x_{n} \right\}$ in $X$ has a subsequence $\left\{ x_{n_{k}} \right\}$ that converges to a point in $X$.

Explanation

In general, in topological spaces, compactness and sequential compactness are independent. For example, there are spaces that are compact but not sequentially compact, and vice versa, spaces that are sequentially compact but not compact.

In metric spaces, compactness is equivalent to sequential compactness.

Theorem

If a space is sequentially compact, it is countably compact.