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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $U \\subseteq \\mathbb{R}^n$ be open and [[connected|0R37DFJKFW]], for the Euclidean topology.\nThen $U$ is [[path-connected|0K0GE6BCYW]] or empty.\n\n## Remarks\n\nProof: fix $x \\in U$, and let $P$ be the set of points of $U$ that a path in $U$ joins to $x$. Each [[open ball|0DC3MPR7C9]] in $U$ is convex, so $P$ and $U \\setminus P$ are both open. As $U$ is connected and $x \\in P$, we get $P = U$.\nThe proof works in each locally path-connected space, for example in each open subset of a normed space.\nIn fact, any two points of $U$ can be joined by a path made of finitely many line segments.\n\n## Explanations\n\n### L1 human\n\nFor open sets in $\\mathbb{R}^n$, being in one piece and being walkable are the same.\nThe bad behaviour of the topologist's sine curve cannot happen in an open set, because each point has a small ball around it.\n",
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