Are there any reasonable conditions one may impose on a path connected space such that it implies that it is locally path connected?
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2026-03-28 10:03:29.1774692209
When does path connectedness imply local path connectedness?
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Unfortunately, this doesn’t hold even for so nice spaces as compact subsets of the plane. To construct such a space $X$ consider a convergent sequence $X_0=\{(0,0)\}\cup \{(1/n,0):n\in\Bbb N\}$ and let $X$ consists of all segments connecting a point $(1,0)$ with points of $X_0$. Then the space $X$ is pathwise connected but not locally pathwise connected at a point $(0,0)$.
On the other hand, local pathwise connectedness is preserved by some operations with topological spaces, see the following exercise from Ryszard Engelkng’s “General topology”.