I'm looking for a topological space locally connected but not connected and not locally path connected. Is there any space with this properties? I haven't been able to find one
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Does this question help? https://math.stackexchange.com/questions/843173/path-connected-and-locally-connected-space-that-is-not-locally-path-connected Space 4 is what you're looking for I believe – Noel Lundström Apr 08 '20 at 16:58
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Especially this answer for the long circle. – Henno Brandsma Apr 08 '20 at 17:03