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I am asking the question in a purely topological setting; a zero-set of some topological space $X$ is a subset which can be realized as the counterimage of a single point through a continuous real-valued function.

Is there any workable characterization of zero-sets which talks only about $X$ and does not deal with other spaces? Of course, if $X$ is perfectly normal, such a characterization is well-known: zero-sets are exactly closed sets. Is there any characterization when $X$ is Tychonoff? I guess that, in the general case of spaces satisfying no special separation axiom, a nice characterization does not exist (of course, "nice" is a subjective notion).

The question is similar in spirit to a remark found in Engelking General Topology: The definition of $T _{3\frac{1}{2}}$-spaces is significantly different from definitions of other classes of spaces studied in this section. It is an external definition, in which we assume the existence of some objects external to the space under consideration (in this case we assume the existence of the interval $I$), in distinction to internal definitions, in which only objects internal to the space under consideration are used... An internal characterization of $T _{3\frac{1}{2}}$-spaces is given in Exercise 1.5.G; the reader should note how complicated and less natural it is.

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    $\begingroup$ A subset of a normal space is a zero-set if and only if it is a closed $G_\delta$ subset. $\endgroup$ Jan 29, 2016 at 13:18
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    $\begingroup$ For Tychonoff spaces all I know is: compact $G_\delta \Rightarrow$ zero-set $\Rightarrow$ closed $G_\delta$. $\endgroup$ Jan 29, 2016 at 13:23

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A simple such characterization exists when one is working with proximity spaces instead of topological spaces. Suppose that $(X,\delta)$ is a proximity space with complete containment relation $\ll$. Then we say that a subset $Z\subseteq X$ is a proximally zero set if there is some proximity map $f:(X,\delta)\rightarrow[0,1]$ such that $Z=f^{-1}[\{0\}]$. On the other hand, the proximally zero sets are precisely the sets of the form $\bigcap_{n\in\omega}A_{n}$ for some sequence $(A_{n})_{n\in\omega}$ such that $\ldots A_{n+1}\ll A_{n}\ll A_{n-1}\ll\ldots\ll A_{0}$. Equivalently, a zero $A$ is a cozero set if and only if $A=\bigcup_{n\in\omega}A_{n}$ for some sequence $(A_{n})_{n\in\omega}$ with $A_{n}\ll A$ for all $n$. This characterization of proximally zero sets was first explicitly spelled out in my paper 2. Since in a completely regular space, a set is a cozero set if and only if it is a proximally cozero set in some compatible proximity, one can define a cozero set to be a set of the form $\bigcup_{n\in\omega}A_{n}$ such that there is a compatible proximity $\ll$ such that $A_{n}\ll A$ for all $n$.

Another similar internal characterization of zero sets is slightly more complicated and is given in 1. A collection $\mathcal{U}$ of open sets is a completely regular family if whenever $U\in\mathcal{U}$ there are sequences $(U_{n})_{n\in\omega},(V_{n})_{n\in\omega}$ of elements in $\mathcal{U}$ so that $U=\bigcup_{n\in\omega}U_{n}$ and $U_{n}\subseteq V_{n}^{c}\subseteq U$. In 1, it is shown that a set $U$ is a cozero set if and only if $U$ belongs to some completely regular family.

Also, I claim that one can also characterize the cozero sets internally in terms of a way-below relation. Suppose that $(X,\mathcal{T})$ is a topological space. Then define $U\prec V$ if and only if $\overline{U}\subseteq V$. Define $U\ll V$ if and only if there is some $(U_{r})_{r\in[0,1]\cap\mathbb{Q}}$ such that $U=U_{0},V=U_{1}$ and where $U_{r}\prec U_{s}$ whenever $r<s$. The following are equivalent:

  1. $U$ is a cozero set.

  2. There is a sequence $(U_{n})_{n\in\omega}$ of open sets with $U_{0}\ll U_{1}\ll U_{2}\ll\ldots U_{n}\ldots$ and where $U=\bigcup_{n\in\omega}U_{n}$

  3. There exists some sequence $(U_{r})_{r\in[0,1)\cap\mathbb{Q}}$ of open sets with $U_{r}\prec U_{s}$ whenever $r<s$ and where $U=\bigcup_{r\in[0,1)\cap\mathbb{Q}}U_{r}$.

This characterization of cozero sets which does not refer to continuous functions is very useful in point-free topology since this characterization can easily be generalized to a point-free setting, and in point-free topology it is usually more convenient to work with open sets than it is to work with the pointfree analogue continuous real-valued functions. Hence, in point-free topology it is more convenient to define the cozero sets in terms of $\prec$ and $\ll$ than it is to define the cozero sets in terms of frame homomorphisms from the frame of real numbers $(\mathbb{R},\mathcal{T})$ to your original frame $L$.

If $X$ is a completely regular space, then the complete containment relation $\ll$ is completely below relation for the finest proximity compatible with the topology on $X$.

1Short proof of an internal characterization of complete regularity. Harald Brandenburg and Adam Mysior. Canad. Math. Bull. 27(1984), 461-462

2A generalization of the notion of a P-space to proximity spaces. J. Van Name Volume 42, 2013. Pages 357–365

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  • $\begingroup$ The results might have applications also to "classical" general topology. Using Scholar, I have found that Brandenburg and Mysior's characterization has been used in M. Hušek, A. Pulgarín, On characterizing Riesz spaces C(X) without Yosida representation, Positivity, Volume 17, Issue 3 , pp 515-524. I expect that all the results you mention will have further applications. $\endgroup$ Jan 30, 2016 at 22:22

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