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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nThe space $X$ is *locally path-connected* if the following holds.\nFor every point $x \\in X$ and every [[open set|02BH3MCBQB]] $U$ containing $x$, there is a [[path-connected|0WB6DH499T]] open set $V$ with $x \\in V \\subseteq U$.\n\n## Remarks\n\nExample: every open subset $W$ of $\\R^n$ is locally path-connected.\nFor $x \\in U \\subseteq W$ with $U$ open, some open ball around $x$ lies in $U$, and a ball is path-connected by straight segments.\n\nNon-example: the [[comb space|A2Q4RXD162]] is not locally path-connected.\nSmall open sets around $(0, 1)$ contain points of the teeth $\\set{1/n} \\times [0, 1]$, and these cannot be joined to $(0, 1)$ by a path inside the small set.\n\nEvery locally path-connected space is [[locally connected|FDVAHRJKF3]], because path-connected sets are connected.\nThe converse fails in general.\n",
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