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  "body": "\n## Summary\n\nA locally path-connected space is a [[topological space|0CQHFPGJ45]] in which every point has arbitrarily small [[path-connected|0WB6DH499T]] open neighbourhoods.\nEvery locally path-connected space is [[locally connected|FDVAHRJKF3]].\nIn a locally path-connected space, the path components and the connected components coincide.\nSo a connected, locally path-connected space is path-connected, a fact used throughout covering space theory.\n\n## Explanations\n\n### L1 human\n\nNear any point of such a space, a small open piece can be chosen in which every two points are joined by a path inside the piece.\nOpen subsets of $\\R^n$ have this property, because small balls are convex.\nThe [[comb space|A2Q4RXD162]] fails it: near a point on the left edge high up, nearby points on the teeth cannot be joined by short paths.\n",
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