Concerning nearly metrizable spaces (Q2871993)

The authors introduce the pseudo-embedding map \(f:X\to Y\) as a continuous and injective map such that the image \(f(A)\) of every regular open set \(A\) in \(X\) is open and then they call a space \(X\) nearly metrizable if there exists a pseudo-embedding map \(f\) from \(X\) to a metrizable space \(Y\). A number of theorems analogous to classical metrization theory are obtained. For example, it is proved that a space is nearly metrizable if and only if it is almost \(T_3\) and has a \(\sigma\)-locally finite pseudo-base.