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  "body": "\n## Statement\n\nLet $X$ be a [[path-connected space|0K0GE6BCYW]].\nThen $X$ is [[connected|0D939CMME7]].\n\n## Remarks\n\nProof: suppose open sets $U$ and $V$ split $X$, with $x \\in U$ and $y \\in V$. Take a path $\\gamma$ from $x$ to $y$. Then $\\gamma([0, 1])$ is connected, by [[continuous image of a connected set is connected|0EPD87YVG8]] and [[connectedness of intervals|0B9ZQFMQNV]]. But $U$ and $V$ split it, a contradiction.\nThe converse fails, as the [[topologist's sine curve|0K3T4A659R]] shows. It holds for [[open subsets of R^n|0MXN7AFBPE]].\n\n## Explanations\n\n### L1 human\n\nIf you can walk between any two points, the space cannot be split into two separate open parts.\n",
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