We say that $X$ is locally connected at $x$ if for every open set $V$ containing $x$ there exists a connected, open set $U$ with $x\in U\subseteq V$. The space $X$ is said to be locally connected if it is locally connected at $x$ for all $x\in X$. Now let $X$ be a compact $T_0$ topological space whose connected components are locally connected. Is there any characterization for such a space?
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1$\begingroup$ Could you give some hints at the kind of characterization you are looking for? What kind of property would you relate that to? It seems such a basic property that characterizations might be mere rephrasings, would you be interested in sufficient or necessary conditions? $\endgroup$– Benoît KloecknerCommented Mar 27, 2021 at 12:11
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$\begingroup$ @Benoît Kloeckner: I am looking for some sufficient and necessary conditions for this property. $\endgroup$– AlbertoCommented Mar 27, 2021 at 16:19
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$\begingroup$ If the compact space is metrizable, then one can try to characterize such spaces as images under some special maps of the product of the Cantor set and the interval. $\endgroup$– Taras BanakhCommented Mar 31, 2021 at 17:09
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