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.
