On the class of \(\omega _{1}\)-metrizable spaces whose product with every paracompact space is paracompact (Q2502973)

Let \(\mathcal P\) denote the class of \(\omega_1\)-metrizable spaces whose product with every paracompact space is paracompact. The paper contains nice results and interesting constructions in connection with the class \(\mathcal P\). It is shown that a space \(X\) belongs to \(\mathcal P\) if and only if its product with every paracompact space is normal; there are a Lindelöf \(P\)-space \(X\) with \(w(X) =\omega_1\) and a paracompact space \(Y\) such that \(X\times Y\) is not normal. If \(X\in \mathcal P\), then \(X^\omega\) is paracompact. Related questions are considered and some open problems are posed.