On perfectly normal dense subspaces of products
It is shown that whenever \(X\) is a separable metric space with more than one point, then \(X^{2^c}\) has a perfectly normal dense subspace which is not collectionwise Hausdorff. This is an interesting answer to a question of \textit{A. V. Arkhangel'skij}, who has asked in [Topological function spaces. Transl. from the Russian by R. A. M. Hoksbergen. Mathematics and its Applications, Soviet Series. 78. Dordrecht etc.: Kluwer Academic Publishers. ix, 205 p. (1992; Zbl 0758.46026)] whether every normal dense subspace of a product of separable metric spaces is collectionwise normal.
scientific article; zbMATH DE number 5123621