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  "body": "\n## Counterexample\n\nLet $C$ be the [[Cantor set|0BXNDZFQME]] on the segment $[0, 1] \\times \\set{0}$ of the plane, and let $p = (1/2, 1/2)$.\nLet $E \\subseteq C$ be the countable set of end points of the open intervals removed in the construction of $C$.\nFor $c \\in C$, let $L(c)$ be the closed segment from $(c, 0)$ to $p$.\nLet $X_c$ be the set of points of $L(c)$ whose second coordinate is rational if $c \\in E$, and irrational if $c \\in C \\setminus E$.\nThe *Knaster–Kuratowski fan* is $F = \\bigcup_{c \\in C} X_c \\cup \\set{p}$, with the topology of a subspace of the plane.\nThen $F$ is [[connected|0V032VWTHD]], but $F \\setminus \\set{p}$ is [[totally disconnected|0RJSBR9GW1]].\nSo removing one point from a connected space can leave no connected subset with more than one point.\n\n## Remarks\n\nFor two points $c \u003c c'$ of $C$, pick $t$ in a removed interval between them.\nThe line through $p$ and $(t, 0)$ misses $F \\setminus \\set{p}$, and the two open half-planes it bounds split $F \\setminus \\set{p}$ into two open pieces.\nSo a connected subset of $F \\setminus \\set{p}$ lies in a single set $X_c$.\nEach $X_c$ is homeomorphic to the rational or to the irrational points of an interval, which is totally disconnected.\nThe proof that $F$ is connected is harder; it uses the [[Baire category theorem|0A7KGAYHGV]] on the Cantor set.\nThe point $p$ is called a dispersion point of $F$.\nThe space is also called Cantor's leaky tent or Cantor's teepee.\n\n## Explanations\n\n### L1 human\n\nPicture a tent whose poles are the segments from the points of the Cantor set up to a common top.\nEach pole is full of holes, keeping only points at rational heights or only at irrational heights.\nThe whole tent hangs together in one piece, but if one removes the top point, it falls apart into dust: no two points stay in one connected piece.\n",
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