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Let š¯‘‹ be a separable metrizable space. An intersection of clopen subsets of $X$ is called a C-set.

Question. If each singleton of $X$ is a C-set, $A\subseteq X$ is closed and countable, $x\in X\setminus A,\,$ and $A\cup \{x\}$ is a C-set, then is there a clopen subset of $X$ containing $A$ and missing $x$?

This question may be connected to Is there an almost strongly zero-dimensional space which is not strongly zero-dimensional and Do $G_\delta$-measurable maps preserve dimension?.

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    $\begingroup$ @WlodAA You are correct I wanted $x\in X\setminus A$ (thanks for edit). But the answer is not so clear to me... $\endgroup$
    – D.S. Lipham
    Commented Dec 20, 2019 at 18:18
  • $\begingroup$ I was mistaken. (I'll remove a couple of my comments that used to be above). $\endgroup$
    – Wlod AA
    Commented Dec 20, 2019 at 19:33

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