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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CNQXVBHCX]] and let $K \\subseteq X$.\nThe 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 $K$.\nThe space $X$ is sequentially compact when $X$ itself is a sequentially compact set.\n\n## Remarks\n\nHere 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|06JYPJ4WHH]].\nThe 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.\nFor metric spaces, [[compactness and sequential compactness agree|0K8SGQSV7G]]. In general topological spaces, neither notion implies the other.\n\n## Explanations\n\n### L1 human\n\nIn a sequentially compact set, a sequence cannot escape: some part of it always settles down to a point of the set.\nIn $(0, 1]$, the sequence $1/n$ tries to escape to $0$, which is outside the set. So $(0, 1]$ is not sequentially compact.\n",
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