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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nThe space $X$ is *extremally disconnected* if for every [[open set|02BH3MCBQB]] $U \\subseteq X$, the [[closure|0QEAJB91GZ]] $\\overline{U}$ is open in $X$.\n\n## Remarks\n\nExample: a discrete space is extremally disconnected.\nEvery subset is open, so in particular every closure is open.\n\nNon-example: $\\R$ with the usual topology is not extremally disconnected.\nThe set $(0, 1)$ is open, and its closure $[0, 1]$ is not open.\n\nAn equivalent condition: for all disjoint open sets $U, V \\subseteq X$, the closures $\\overline{U}$ and $\\overline{V}$ are disjoint.\nSome authors also require $X$ to be Hausdorff.\nAn extremally disconnected Hausdorff space is [[totally disconnected|0RJSBR9GW1]].\n",
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