Sequential compactness

Definition

Let $X$ be a topological space and let $K \subseteq X$. The set $K$ is sequentially compact when each sequence $(x_n)$ of points of $K$ has a subsequence $(x_{n_k})$ that converges to a point of $K$. The space $X$ is sequentially compact when $X$ itself is a sequentially compact set.

Remarks

Here a sequence $(y_k)$ converges to $y$ when each open set that contains $y$ contains $y_k$ for all large $k$. In a metric space, this is the usual convergence. The interval $[0, 1]$ is sequentially compact by the Bolzano–Weierstrass theorem. The real line is not: the sequence $x_n = n$ has no convergent subsequence. For metric spaces, compactness and sequential compactness agree. In general topological spaces, neither notion implies the other.

Explanations

L1 human

In a sequentially compact set, a sequence cannot escape: some part of it always settles down to a point of the set. In $(0, 1]$, the sequence $1/n$ tries to escape to $0$, which is outside the set. So $(0, 1]$ is not sequentially compact.