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  "body": "\n## Example\n\nLet $X$ be a set and let $p \\in X$.\nThe *particular point topology* on $X$ declares a set $U \\subseteq X$ [[open|02BH3MCBQB]] if $U = \\emptyset$ or $p \\in U$.\nThis is a topology: $\\emptyset$ and $X$ are open.\nA union of sets that contain $p$ contains $p$, and so does a finite intersection of them.\nSo $X$ is a [[topological space|0CQHFPGJ45]].\n\n- $X$ is a [[T0 space|07SG12TQH3]]: let $x \\ne y$, and name them so that $x \\ne p$.\n  Then $\\set{p, y}$ is open, contains $y$, and does not contain $x$.\n- If $X$ has at least two points, $X$ is not a [[T1 space|0NW6CH38YR]]: every open set that contains a point $x \\ne p$ also contains $p$.\n- $X$ is [[connected|0V032VWTHD]]: any two nonempty open sets meet at $p$, so $X$ is not a union of two disjoint nonempty open sets.\n- The set $\\set{p}$ is [[dense|0BSJSX7ME2]], since every nonempty open set contains $p$.\n- $X$ is [[compact|0TVJZYB6SH]] if and only if $X$ is finite.\n  Indeed, the open sets $\\set{p, x}$ for $x \\in X$ cover $X$, and a finite subcover exists only when $X$ is finite.\n\n## Explanations\n\n### L1 human\n\nIn this topology one point $p$ is special: a nonempty set is open exactly when it contains $p$.\nSo $p$ is \"close\" to every other point, and every small open set around any point must reach $p$.\nThis makes the space connected and lets the single point $p$ be dense.\nIt is a standard source of counterexamples in general topology.\n",
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