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