A remark on Katetov's theorem (Q1813771)

The author studies pseudonormal product spaces. A \(T_ 1\) space is pseudonormal if every two disjoint closed subsets one of which is countable have disjoint open neighbourhoods. It is said to be \(F_ \sigma\)-pseudonormal if each of its \(F_ \sigma\)-subspaces is pseudonormal. A space \(X\) is pointwise \(F_ \sigma\)-pseudonormal if \(X- \{x\}\) is \(F_ \sigma\)-pseudonormal for each \(x\in X\). The results include the following facts: Let \(X\) be a space containing a countable subset which is not closed. Then (1) if \(X\times Y\) is hereditarily pseudonormal, then each countable closed subset of \(Y\) is a \(G_ \delta\)-set, and (2) if \(X\times Y\) is pointwise \(F_ \sigma\)- pseudonormal, then each one-point subset of \(Y\) is a \(G_ \delta\)-set.