Normality in products with a countable factor
This paper begins by quickly establishing that if \(X \times Y\) is normal for some space \(Y\) with a countable non-discrete subspace, then \(X\) is not Dowker (= normal but not countably paracompact). The author then turns his attention to investigating which Dowker spaces \(X\) have \(X^2\) normal. Rudin's original Dowker space (which has size \((\aleph_{\omega})^{\omega}\)) does have a normal square. Are there any small Dowker spaces with normal square? The answer in this paper is: yes, under \(\diamondsuit^*_S\) there is such a space, with size \(\omega_2\). The author then investigates spaces \(X\) in which \(X \times Y\) is not normal for some countable \(Y\). (Purisch and Rudin gave conditions in which a normal space \(X\) cannot have this property, for example, a linearly ordered space has normal product with any countable space.) He proves a number of theorems, including: 1. under CH there is a countably compact normal space \(X\) and a countable regular space \(Y\) with \(X \times Y\) not normal; 2. Under \(\diamondsuit\) there is an Ostaszewski space \(X\) and a countable regular space \(Y\) with \(X \times Y\) not normal; 3. the non-metrizable normal Moore space constructed from a \(Q\)-set in \(2^{\omega}\) has non-normal product with a countable space.
scientific article; zbMATH DE number 5710099