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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nA subset $P \\subseteq X$ is *perfect* if $P$ is [[closed|0ZCK8DQYN1]] in $X$ and $P$ has no [[isolated points|0A1YXSKB48]].\nEquivalently, $P = P'$, where $P'$ is the [[derived set|8ZAPAPYJFG]] of $P$.\n\n## Remarks\n\nThe two forms agree because $P$ is closed if and only if $P' \\subseteq P$, and $P$ has no isolated points if and only if $P \\subseteq P'$.\n\nExample: $[0, 1]$ is closed in $\\R$, and every $x \\in [0, 1]$ is a limit of other points of $[0, 1]$, so $[0, 1]$ is perfect.\nThe [[Cantor set|0BXNDZFQME]] is also perfect.\n\nNon-example: $P = \\set{0} \\cup \\set{1/n : n \\ge 1}$ is closed in $\\R$, but $1$ is an isolated point of $P$, so $P$ is not perfect.\n\nNon-example: $\\Q$ has no isolated points, but it is not closed in $\\R$, so it is not perfect.\n\nEvery nonempty perfect subset of $\\R$ is uncountable.\n",
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