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  "body": "\n## Example\n\nLet $X$ be a set and let $p \\in X$.\nThe *excluded point topology* on $X$ declares a set $U \\subseteq X$ [[open|02BH3MCBQB]] if $U = X$ or $p \\notin U$.\nThis is a topology: $\\emptyset$ and $X$ are open.\nA union of open sets is either $X$ or avoids $p$, and so is a finite intersection of open sets.\nSo $X$ is a [[topological space|0CQHFPGJ45]].\n\n- The [[closed sets|0ZCK8DQYN1]] are $\\emptyset$ and the sets that contain $p$.\n- $X$ is [[compact|0TVJZYB6SH]]: the only open set that contains $p$ is $X$, so every open cover has $X$ as a member.\n  Then $\\set{X}$ is a finite subcover.\n- $X$ is [[connected|0V032VWTHD]]: a clopen set that contains $p$ is open, hence equal to $X$.\n  A clopen set that misses $p$ is closed, hence empty.\n- $X$ is a [[T0 space|07SG12TQH3]]: for $x \\ne p$, the open set $\\set{x}$ contains $x$ but not $p$.\n  For distinct $x, y \\ne p$, the open set $\\set{x}$ separates them.\n- If $X$ has at least two points, $X$ is not a [[T1 space|0NW6CH38YR]], since every open set that contains $p$ contains all points.\n\nThe closed sets of this topology are exactly the open sets of the [[particular point topology|4VSZ22TX4B]] at $p$.\n\n## Explanations\n\n### L1 human\n\nHere one point $p$ is kept out: a set is open when it avoids $p$, and the whole space is open too.\nEvery point other than $p$ is open by itself, so away from $p$ the space looks discrete.\nThe point $p$ can only be surrounded by the whole space, which is why one member of any open cover already covers the space, and the space is compact.\nIt is the mirror image of the particular point topology.\n",
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