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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nA subset $A \\subseteq X$ is *clopen* if $A$ is [[open|02BH3MCBQB]] in $X$ and $A$ is [[closed|0ZCK8DQYN1]] in $X$.\nEquivalently, $A$ and its complement $X \\setminus A$ are both open.\n\n## Remarks\n\nExample: let $X = [0, 1] \\cup [2, 3]$ with the topology from $\\R$.\nThe set $A = [0, 1]$ equals $X \\cap (-1, 3/2)$, so it is open in $X$.\nIts complement $[2, 3]$ equals $X \\cap (3/2, 4)$, so it is open as well, and $A$ is closed.\n\nExample: in the rationals $\\Q$ with the topology from $\\R$, the set $\\set{q \\in \\Q : q \u003c \\sqrt{2}}$ is clopen, since $\\sqrt{2} \\notin \\Q$.\n\nNon-example: in $\\R$, the set $[0, 1)$ is not open, because no interval around $0$ lies inside it.\nIt is not closed either.\n\nIn every space, $\\emptyset$ and $X$ are clopen.\nA space $X$ is [[connected|0V032VWTHD]] if and only if these are its only clopen subsets.\n",
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    "name": "Clopen set",
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