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Let $X$ be a compact $T_0$ topological space such that its closed and connected subsets are path connected. Is there any characterization for such a space?

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    $\begingroup$ Not an answer. Consider the set $C$ in the plane defined as the union of the vertical segment $S:=\{0\}\times [0,1]$ with the graph of the function $$f:(0,1]\to\mathbb{R},$$ $f(x)=\sin(1/x)$. The set $C$ is closed, connected and I do not believe that it is path connected (Better check my claim). If my claim is correct, then the spaces you are enquiring are pretty weird: no closed subset of such a space can be homeomorphic to a closed cube of dimension $\geq 2$. With the caveat that my claim could be horribly wrong. $\endgroup$ Aug 13, 2020 at 18:49
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    $\begingroup$ @LiviuNicolaescu Your space is not path connected. $\endgroup$ Aug 13, 2020 at 19:08
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    $\begingroup$ @LiviuNicolaescu Unless I'm mistaken, the interval $[0,1]$ has this property. Of course every compact totally disconnected space is also an example (for trivial reasons). $\endgroup$ Aug 13, 2020 at 21:09
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    $\begingroup$ Wouldn't 1-dimensional CW complexes work likewise? $\endgroup$ Aug 13, 2020 at 21:16
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    $\begingroup$ @DenisNardin More generally, every hereditarily locally connected (i.e. every connected subspace is locally connected) Polish space has this property. $\endgroup$ Aug 13, 2020 at 21:16

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