On metrizability of paracompact \(p\)-spaces (Q1358443)

Paracompact \(p\)-spaces can be defined as topological spaces that are preimages of metric spaces under perfect mappings. The author obtains the following Theorem 2. A paracompact \(p\)-space \(X\) is metrizable if and only if there exists a continuous function \(\varphi:X^3\smallsetminus \Delta\to[-1,1]\) (where \(\Delta\) is the diagonal), satisfying the condition \(\varphi(x,x,y)\neq \varphi(x,y,y)\) for all \(x,y\in X\), \(x\neq y\). Theorem 2 yields the Corollary. For a paracompact \(p\)-space \(X\) the following conditions are equivalent: (1) the space \(X\) is metrizable; (2) \(X^2\smallsetminus \Delta\) admits a \(\sigma\)-locally finite (in \(X^2\smallsetminus \Delta)\) functionally open (in \(X^2\)) covering.