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Nov 20, 2010 at 19:20 comment added Vít Tuček Of course. I should think at least three times before I post an answer. I was thinking about nonlocal case, i.e. every open connected subset is path connected.
Nov 20, 2010 at 17:49 comment added Nate Eldredge Easier than that: an open set is always locally path connected, because it contains balls centered at every point.
Nov 20, 2010 at 17:21 history answered Vít Tuček CC BY-SA 2.5