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  "body": "\n## Definition\n\nLet $X$ be a [[Polish space|584GXXY73M]] and let $A \\subseteq X$.\nThe set $A$ has the *perfect set property* if $A$ is [[countable|01TQB88H4C]] or $A$ contains a nonempty [[perfect set|628242KN0T]].\nHere a perfect set is a closed subset of $X$ with no isolated points.\nA class $\\Gamma$ of subsets of Polish spaces has the perfect set property if every set in $\\Gamma$ has it.\n\n## Remarks\n\nExample: the Cantor set $C \\subseteq \\R$ has the property, because it is nonempty, closed and has no isolated points, so it contains the perfect set $C$ itself.\n\nNon-example: under the axiom of choice there is a Bernstein set $B \\subseteq \\R$, a set such that both $B$ and $\\R \\setminus B$ meet every uncountable closed set.\nThen $B$ is uncountable, but $B$ contains no nonempty perfect set $P$, because $P$ is an uncountable closed set and so meets $\\R \\setminus B$.\n\nEvery closed set has the property by the Cantor–Bendixson theorem, and so does every Borel set and every analytic set.\n",
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