Why is Urysohn's metrization theorem so much more "popular" than the Nagata–Smirnov metrization theorem?

Question

The Urysohn's metrization theorem is much better known than the Nagata–Smirnov metrization theorem. However, the former only gives a sufficient condition for metrizability whereas the latter gives a necessary and sufficient condition.

I would imagine that giving a full characterization of when a topological space is metrizable should be a huge deal, perhaps something deserving of a fields medal even. Yet it seems as though most people don't really think this way, which is why this result isn't really particularly well-known.

What's the reason for this?

Answer 1

Historically, it was an important problem to find a purely topological characterisation of a space $X$ to be metrisable, without a reference to an external object like $\mathbb{R}$. Urysohn was one of the first to work on this and he proved some important theorems on embedability using continuous functions on normal spaces (Urysohn's lemma). He could then characterise the separable metric spaces as those that could be embedded into $[0,1]^\mathbb{N}$ (using Urysohn functions and a countable base to reduce the number of dimensions in the cube to countable). But the general problem was still open. But Urysohn's special case turned out to be already useful to find metrisable spaces in analysis, e.g. Moreover, it can be proved easily once you covered normal spaces and Urysohn's lemma, so it occurs in many text books.

Later in the 40's and 50's people started studying open covers and refinements, paracompactness was defined and shown to be equivalent to being "fully normal". Metric spaces were proved to be paracompact that way, but there were also many non-metric paracompact spaces so one needed a strengthening of paracompactness, and this was found to be having a $\sigma$-locally finite base (which by results known at that time immediately implied paracompactness) and which was then shown to be equivalent to metrisability by both Smirnov (in Russia) and Nagata (in Japan), independent of each other. Around the same time Bing (in the US) shows that having a $\sigma$-discrete base (plus regular $T_0$, which we also need in Nagata-Smirnov) was also necessary and sufficient and used this to prove that a space $X$ is metrisable iff it is a developable regular Hausdorff space which is collectionwise normal (a collectionwise normal Moore space). Other characterisations were also found, using special bases most of the time. But most of that theory, though interesting, is quite extensive to fully cover in most courses, (Munkres does Nagata-Smirnov as an optional chapter, e.g.) and the problem was to show that metrisability is "intrinsically characterisable", not as a practical method to discover metric spaces, or prove concrete spaces to be metrisable.

It has been used to characterise metrisability in special classes of spaces, like (generalised) ordered spaces. It maybe deserves to be better known, as other metrisation theorems too, but it's too "deep" into general topology problems, and not as practical as Urysohn's theorem, which has an easier to check condition.

Answer 2

I think the reason for the focus on the Urysohn metrization theorem is really Urysohn's lemma, which is extremely useful and widely applicable, and the Urysohn metrization theorem is just one easy application that one might as well prove anyway. Essentially, the Nagata-Smirnov metrization theorem also uses the Urysohn lemma. Characterizing metric spaces, while interesting, is less widely applicable, and the new ideas that were required are less of a breakthrough.

In reality, most modern topologists care very little about weird spaces that they need metrization theorems to prove are metrizable. Certainly there are some who do still study these areas, but mostly topologists are focused on algebraic topology where in the sense of general topology the spaces are "boring," for example compact metric spaces, very often manifolds.