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math typo fixed
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edit approved Dec 20, 2019 at 6:54
Wlod AA
edit approved Dec 20, 2019 at 6:54
Wlod AA
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Let 𝑋 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, $x\in X$,$x\in X\setminus A,\,$ 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?.

Let 𝑋 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, $x\in X$, 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?.

Let 𝑋 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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D.S. Lipham
D.S. Lipham
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Separating a countable closed set from a point

Let 𝑋 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, $x\in X$, 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?.