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Let $X$ be a compact non-Hausdorff topological space. I am looking for a characterization for the following property on $X$:

Property: For each non-empty closed subset $C$ of $X$ there exists a partition $C=C_1\bigsqcup\cdots\bigsqcup C_n$ of $C$ into closed subsets $C_i$ of $X$ and there exists a non-empty open set $U_C$ such that for each $i$, $U_C\cap C_i$ is contained in a connected component of $C_i$.

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  • $\begingroup$ $U_C$ should be nonempty, right? $\endgroup$
    – erz
    Commented Mar 14, 2020 at 20:22
  • $\begingroup$ Yes, $U_C$ should be non-empty. Thank you for your comment. $\endgroup$
    – Alexander
    Commented Mar 14, 2020 at 20:42
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    $\begingroup$ Perhaps some motivation for this question would be helpful? I cannot quite see what this property is about.. $\endgroup$
    – erz
    Commented Mar 14, 2020 at 20:54
  • $\begingroup$ For compact Hausdorff spaces this property is equivalent to the scatteredness (= each non-empty closed subspace has an isolated point). For compact (not necessarily Hausdorff) spaces, the scatteredness implies the property in OP. An example of compact non-Hausdorff non-scattered space with the property is the real line endowed with the cofinite topology. $\endgroup$
    – Taras Banakh
    Commented Mar 28, 2020 at 16:15

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