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  "body": "\n## Definition\n\nA [[topological space|0CQHFPGJ45]] $X$ is *limit point compact* when every infinite subset $A \\subseteq X$ has a [[limit point|0E3X7TXS67]] in $X$.\nHere $x \\in X$ is a limit point of $A$ when every neighbourhood of $x$ contains a point of $A$ other than $x$.\n\n## Remarks\n\nEvery [[compact|0TVJZYB6SH]] space is limit point compact, and so is every [[countably compact|S3N3CHZV89]] space.\nFor [[T1 spaces|0NW6CH38YR]], limit point compactness is equivalent to countable compactness.\nFor [[metric spaces|0VKD1TP7F3]], it is equivalent to compactness and to sequential compactness.\nOther names in use are weakly countably compact and Fréchet compact.\n\nExample: $[0, 1]$ is limit point compact.\nAn infinite subset contains a sequence of distinct points, which has a convergent subsequence by the Bolzano–Weierstrass theorem, and its limit is a limit point of the set.\n\nNon-example: $\\R$ is not limit point compact.\nThe infinite set $\\Z$ has no limit point, since each $x \\in \\R$ has a neighbourhood $(x - 1/2, x + 1/2)$ that contains at most one integer.\n\nWithout the T1 condition, limit point compact need not imply countably compact.\nLet $X = \\N \\times \\set{0, 1}$, where $\\N$ is discrete and $\\set{0, 1}$ has the trivial topology.\nEach nonempty subset has a limit point, since $(n, 1 - i)$ is a limit point of any set that contains $(n, i)$.\nBut the countable open cover by the sets $\\set{n} \\times \\set{0, 1}$ has no finite subcover.\n",
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