Metrization problem

Question

Can the following problem be solved by Nagata Smirnov metrization theorem, which states that

A topological space is metrizable if and only if it is frechet, regular and has a sigma locally finite basis.

It can be easily shown that it is frechet and regular but I can't prove the existence of a sigma locally finite basis.

Answer 1

The original paper for this result (stated on page 140) uses a theorem by Alexandroff and Urysohn quoted on p136 loc.cit. on metrization of Fréchet "voisinage"-spaces (a precursor to our modern topological spaces).

The modern version of this is in Engelking's General Topology as Thm 5.4.9 (the Alexandroff-Urysohn metrisation theorem, "the chronologically earliest metrization theorem"):

A topological space $X$ is metrisable iff it is a $T_0$ space and has a development $\mathcal{W}_1, \mathcal{W}_2, \ldots$ such that for every $i \in \Bbb N$ and any two $W_1,W_2 \in \mathcal{W}_{i+1}$ such that $W_1 \cap W_2 \neq \emptyset$, there exists a $W \in \mathcal{W}_i$ such that $W_1 \cup W_2 \subseteq W$.

This in turn can be derived from Moore's metrisation theorem

(a space $X$ is metrisable iff it is $T_0$ and has a strong development $\mathcal{W}_1, \mathcal{W}_2, \ldots$ (strong means that for every $x$ in $X$ and every open neighbourhood $U$ of $x$ there is some $i \in \Bbb N$ and some open neighbourhood $V$ of $x$ such that $\bigcup \{W \in \mathcal{W}_i\mid W \cap V \neq \emptyset\} \subseteq U$; for a normal development we just ask for $i$ such that $\bigcup \{W \in \mathcal{W}_i\mid x \in W\} \subseteq U$).

Moore's metrisation theorem in turn again can be quickly derived from Bing's metrisation theorem that

$X$ is metrisable iff it's collectionwise normal (including $T_1$) and has a development.

The last one has at least one proof based on Bing-Nagata-Smirnov.

So yes there is a proof based on B-N-S. Read Engelking's book section 5.4, where all this is covered in detail. The notes and exercises to that section are informative (as to chronology, who proved what, and variations of such theorems).