I am quite struck by the similarity between covering charachterization of Paracompactness and Metrizability both in the T1 and T3 topologies.
In particular in the case of T1 topologies we have the following two results due to Arhangelskii
-A T1 space is paracompact if and only if for each open covering there exist locally starring sequence
-A T1 space is metrizable if and only if there exist a locally starring sequence common to each open covering
On the other hand, for T3 topologies we have the following charachterization of paracompact spaces by Michael and the famous Nagata-Smirnov theorem for metrizability
-A T3 space is paracompact if and only if each open covering of Y can be refined to an open covering made up by an at most countable collection of nbd-finite families of open sets.
-A T3 space is metrizable if and only if it has a basis that can be decomposed into an at most countable collection of nbd-finite families.
(Again notice how a a basis can be seen as a common refinement to any open covering)
Then my question: is this just a coincidence? Is there a profound meaning and explanation to the fact that metrizable spaces seem to be paracompact spaces where the characterization property becomes somehow a common one?
Furthermore I noticed that this feature of showing common refinements shows up again in the definition of uniformities through coverings
Best Answer
This is indeed no coincidence; both the metrisation theorems and notions like paracompactness, star-refinements and all sorts of other covering properties were sort of developped in parallel, around the same time. Maybe you can look in the Handbook of the History of General Topology for more exact info. Or read the historical notes in Engelking around the sections on covering properties and metrisation theorems. You'll find many more similarities (both paracompactness and metrisabiltiy are preserved by closed maps, e.g. and are inverse invariants of perfect maps, etc. Spaces like $\omega_1$ and the Sorgenfrey line serve as motivation/delimiting examples. A whole subfield of "generalised metric spaces" has evolved, e.g. those having special bases or strong normality properties (screenable, monotonically normal etc.), and paracompactness nicely fits in there as well. I recommend reading section 5.4 ("Metrization Theorems II"), which fittingly enough sits inside the chapter on paracompactness, and covers Alexandroff's and Arhangel'skij's metrisation theorems within that context; the Bing-Nagata-Smirnov theorem is in the chapter before on metric spaces, and that chapter already hints at theorems that are really more about paracompactness than metrisability. The historical section of 5.4 is a nice overview of different metrisation theorems.
Covers not only play an important role in uniform spaces (there are several equivalent ways to do uniform spaces, the idea of doing it by "uniform covers" came again from the same people that studied paracompactness and other covering properties), but also in homology theories, dimension theory, etc. So those ideas also found their applications there. Around the 1950's this whole idea of working with covers and refinements of different types was "in the air" and many of the basic results we still use have been proven around that time.
Good you noticed the similarities, and no, they're not coincidental but reflect a decades long trend in general topology research.