A few standard results (on metrizability and relative separation strength) without Choice?

Question

I've been going back over some results from Munkres's Topology, and I'm curious about some things. (I originally posted this on M.SE, but I think it is probably a better fit here.)

I know that Choice principles have some connection to the separation axioms (in ZF, at least)--for example:

Locally compact Hausdorff spaces are Baire if and only if the Axiom of Dependent Multiple Choices holds. [Due to Fossy and Morillon, I believe.]

Complete pseudometric spaces are Baire if and only if the Axiom of Dependent Choices holds.

Still, it seems likely that compactness (or the weaker condition of completeness) and "Baireness" may be playing a substantial part, here.

I know that the Urysohn Metrization Theorem--which states that regular, Hausdorff, second-countable spaces are metrizable (or equivalently, that a space is separable and metrizable if and only if it is regular, Hausdorff, and second countable)--holds in ZF, though the Urysohn Lemma--which states that a space is normal Hausdorff if and only if any two disjoint closed sets can be separated by a continuous function--does not. That's a nice result.

First, the usual proof of the Urysohn Lemma (cf. Munkres) uses Dependent Choice, and it has been suggested to me that Dependent Multiple Choices may do the trick. Does anyone know a proof for this?

Second, I wonder what is known about how much Choice is needed to prove the following metrization theorems, or whether any of them are known to be equivalent in ZF:

Nagata-Smirnov Metrization Theorem: A topological space $X$ is metrizable if and only if it is $T_3$ and has a basis that is countably locally finite [= $\sigma$-locally finite base].

Smirnov Metrization Theorem: A topological space is metrizable if and only if it is paracompact Hausdorff and locally metrizable.

Bing Metrization Theorem: A topological space $X$ is metrizable if and only if it is $T_3$ and has a basis that is countably locally discrete [= $\sigma$-discrete base].

Answer 1

• Paul Howard, Kyriakos Keremedis, Jean E. Rubin, Adrienne Stanley, Paracompactness of metric spaces and the axiom of multiple choice [MLQ 46 (2000), 219–232; MR1755811]

• Omar De la Cruz, Eric Hall, Paul Howard, Kyriakos Keremedis, Jean E. Rubin, Metric spaces and the axiom of choice [MLQ 49 (2003), 455–466; MR1998077]

They show that the Nagata–Smirnov and Bing metrization theorems are equivalent to Stone's Theorem (paracompactness of metric spaces) though the backward implications are provable in ZF.

Answer 2

To prove Urysohn's Lemma using Dependent Multiple Choice (DMC), proceed as in the usual proof, using Dependent Choice (DC), with the following modification. At each stage of the usual inductive construction, you need to find some open sets (usually $2^n$ of them at stage $n$) subject to requirements that look like this: You already have open sets $U$ and $V$ with the closure $\overline U$ of $U$ included in $V$, and you want an open $W$ with $\overline U\subseteq W\subseteq \overline W\subseteq V$. Normality of thespace gives you the existence of such a $W$, but you use DC to choose appropriate $W$'s for all the stages together. Using DMC instead of DC, you get, instead of a particular $W$ for given $U$ and $V$, a finite set of such $W$'s. But then you can replace those finitely many $W$'s with their intersection (or their union) to get a single appropriate $W$. That's the essential idea of the proof; one still needs to be careful about some technical details, but, if I remember correctly, they're rather straightforward.