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(All spaces are Hausdorff.)

This question is a variant of my previous question.

Let $X$ be a completely regular space, that is, for every closed set $F\subseteq X$ and $x\not\in F,$ there exists a continuous function $g:X\to [0,1]$ such that $g(F) = \{0\}$ and $g(x) =1.$

Question: For every closed $G_\delta$ set $F\subseteq X,$ does there exist a continuous function $f:X\to [0,1]$ such that $f=0$ on $F$ and $f\neq 0$ outside $F?$

A subset $U\subseteq X$ is a zero set if there exists a continuous function $g:X\to [0,1]$ such that $g^{-1}(\{0\}) = U.$

It is well-known that if $X$ is normal then all closed $G_\delta$ sets are zero sets.

However, I am not sure whether the same holds for completely regular space.

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  • $\begingroup$ I know that $X$ normal implies that all closed $G_\delta$'s are zero-sets, but I believe the reverse does not hold. If it did your question would be useless. Do you have a reference ? "well-known" it is not, it's not mentioned in Engelking, a standard reference for such matters. $\endgroup$ – Henno Brandsma Jan 5 at 14:26
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    $\begingroup$ Note that this can only be true when $F$ is closed and $G_\delta$. You cannot ask this for all closed sets. $\endgroup$ – Henno Brandsma Jan 5 at 14:27
  • $\begingroup$ @HennoBrandsma yes, you are right. I have edited my post. $\endgroup$ – Idonknow Jan 5 at 14:59
  • $\begingroup$ @HennoBrandsma by the way, if the converse holds, why would by question be useless? $\endgroup$ – Idonknow Jan 5 at 15:00
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    $\begingroup$ @PaulFrost that is the direction I agreed with, yes. The OP originally claimed that if all closed $G_\delta$ sets are zero-sets then $X$ is normal. This I doubt the truth of, as said. $\endgroup$ – Henno Brandsma Jan 5 at 18:46
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There is a well-known counterexample. Let $S$ be the Sorgenfrey line, that is the real line endowed with the Sorgenfrey topology (generated by the base consisting of half-intervals $[a,b)$, $a<b$). It is well-known that a product $X=S\times S$ is a Tychonoff but non-normal space (see, for instance, Examples 1.4.4 and 2.3.12 in [Eng]). Let $D=\{(x,-x)\in S\times S: x\in S\}$. Then any subset $Y$ of $D$ is closed in $X$, and we have $2^{|D|}=2^{\frak c}$ such subsets. Moreover, since $Y=\bigcap_{n\in\Bbb N} Y+[0,1/n)\times [0,1/n)$, $Y$ is a $G_\delta$-subset of $X$. On the other hand, let $C$ be a countable dense subset of $X$. By Theorem 2.1.9 in [Eng], each continuous real-valued function $f$ on $X$ is uniquely determined by its restriction on $C$, so there are at most $\frak c^\omega=\frak c$ such functions. Thus most subsets of $D$ are not zero-sets.

References

[Eng] Ryszard Engelking, General Topology, 2nd ed., Heldermann, Berlin, 1989.

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