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One definition of a perfectly normal ($T_6$) space is that any two closed sets $A$ and $B$ can be precisely separated by a function, i.e. there is a continuous function from the whole space to the closed interval $[0,1]$ for which $f(a) = 0\leftrightarrow a\in A$ and $f(b) = 1\leftrightarrow b\in B$. What applications are there for precisely separating closed sets by a function?

The only example I have been able to find so far is the Nagata-Smirnov metrization theorem, which says that a space is metrizable iff it is regular, Hausdorff, and has a countably locally finite basis. The first step is to show that a regular space with a countably locally finite basis is perfectly normal---once that is done you use the basis to grab a bunch of closed sets, invoke perfectly-normal-ness to get a bunch of functions which essentially measure distances from points to those closed sets, add up all the distances in a way that converges to get a pseudometric, and then invoke the Hausdorff property to show that the pseudometric is really a metric. OK, that's a useful application of "precisely separating closed sets by functions," but it doesn't motivate the notion of a perfectly normal space: if you have a perfectly normal Hausdorff space, you still need that countably locally finite basis to construct the metric, so assuming that the space is "perfectly normal" from the start doesn't buy you anything that you couldn't derive by assuming regularity.

I recognize this is a vague question but I still believe it's worth asking. Are there any important applications of precisely separating two closed sets by functions other than the Nagata-Smirnov metrization theorem?

To rephrase: I'm looking for an example of a theorem that:

  • Applies to perfectly normal spaces
  • Does not apply to completely normal spaces
  • Applies to metric spaces (which are perfectly normal) but doesn't require all of the properties of a metric space (i.e. the theorem doesn't require the space to be Hausdorff or to have a countably locally finite basis).

I don't want a characterization like "Perfectly normal spaces are the ones in which all closed sets are $G_\delta$." That's true, but the fact that closed sets are $G_\delta$ is not intrinsically interesting---it's a fact that has value only after we've already decided that perfectly normal spaces are interesting for some other reason, and now we are looking for ways to characterize them.

A.C.
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  • The fact that closed sets are $G_\delta$'s does have applications, though. It leads, for example, to the fact that every finite Borel measure on such a space is regular. https://math.stackexchange.com/a/82048/1210477 – M W Oct 03 '23 at 00:49
  • That's a fair point, and on reading more about $G_\delta$ sets I am coming to accept their value. But apparently there are spaces in which all closed sets are $G_\delta$ but which are not normal (and therefore not perfectly normal), and from what I can tell most (all?) of the interesting theorems related to $G_\delta$ spaces make no use of normality. So: there may be value to making functions which are zero precisely on any given closed set (which is not something I had imagined--thanks for that) but maybe not so much value in functions which "precisely separate" two closed sets. – A.C. Oct 03 '23 at 02:48
  • @A.C. Yes that's true, Moore's plane is an example of such space. The characterization is not that every closed sets has to be $G_\delta$, but it has to be a zero set. Then you recover perfectly normal spaces. – Jakobian Aug 29 '24 at 17:02

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An example of a theorem that applies to perfectly normal spaces and doesn't apply to completely normal spaces is the following one:

Every perfectly normal space is countably paracompact. See the proof here (corollary to theorem 2). Evidently, it applies for metrizable spaces, as they are perfectly normal. But it doesn't require all of the properties of such spaces, it is sufficient for the space to be perfectly normal.

Examples:

The Odd-Even Topology is perfectly normal and it isn't Hausdorff, therefore, it isn't metrizable neither.

Niemytzki's tangent disc topology is $G_{\delta}$, $T_3{\frac{1}{2}}$, but it isn't countably paracompact.

And, here, @Jakobian provides a great article in which it's proved that there exists a $T_5$ (completely normal and $T_1$) space of cardinality $\mathfrak{c}$ and that isn't countably paracompact.

Almanzoris
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    A quick proof that a perfectly normal space is countably paracompact is as follows: Take a countable open cover $U_i$ of $X$, since open sets are cozero sets and the cover is countable we can find a partition of unity $f_i$ with $U = {f_i > 0}$ and from Mather's theorem there further exists a locally finite partition of unity $g_i$ with $V_i = {g_i > 0}\subseteq {f_i > 0}$. – Jakobian Aug 29 '24 at 16:52
  • Thank you for your quick and brilliant explanation, Jakobian. – Almanzoris Aug 29 '24 at 17:17