Dense Subspaces of Product Spaces

Abstract

Unless otherwise specified, all spaces considered here are regular T1-spaces. A space X is called σ-discrete if X is the union of a countable family of discrete subspaces. Arhangel'skii [2] showed that the class of spaces which contain dense σ-discrete subspaces is productive. The fact that the class of spaces which contain dense subspaces of countable pseudocharacter is productive is obtained by Amirdzanov [1]. On the other hand, the class of spaces which contain metrizable spaces as dense subspaces is obviously not productive. As a generalized concept of metrizable spaces there is the concept of σ-spaces [14]. This class of spaces has many similar properties to the class of metrizable spaces. However we will point out a remarkable difference between the class of metrizable spaces and the class of σ-spaces by showing that the class of spaces which contain σ-spaces as dense subspaces is productive.