Partitions of unity and coverings (Q2250124)

Given a set \(S\) let \(l_1(S)\) be the set of functions \(f\in \mathbb{R}^S\) such that \(\|f\|=\sum\{|f(s)|: s\in S\}<\infty\). The authors observe that a partition of unity on a space \(X\) can be considered as a map from \(X\) to the set \(\Delta(S)=\{f\in l_1(S): f(s)\geq0\) for any \(s\in S\) and \(\|f\|=1\}\). For any \(s\in S\) let \(\text{st}(s)=\{f\in l_1(S): f(s)\neq0\}\). The authors show that, using partitions of unity, it is possible to deduce many famous metrizations theorems (in particular, the Kakutani-Birkhoff metrization theorem for groups, Bing's metrization theorem and Alexandroff's metrization criterion) from Dydak's metrization theorem which states that a space \(X\) is metrizable if and only if, for some set \(S\), there exists a continuous partition of unity \(\varphi: X\to l_1(S)\) such that the family \(\{\varphi^{-1}(\text{st}(s)): s\in S\}\) is a base of \(X\).