A note on spaces in which every open set is $z$-embedded

Abstract

Let Oz be the class of topological spaces in which every open set is z-embedded. In this note we prove the following: If Y is a dense subspace of the real line, then the spaces sY and sY — Y are not in Oz. Introduction. A subset S of a topological space X is z-embedded in X if every zero-set in S is the intersection of S with a zero-set in X. (A zero-set is the set of zeros of a real-valued continuous function.) Blair (1) studied the class Oz of topological spaces in which every open set is z-embedded. This class includes all perfectly normal spaces, all extremally disconnected spaces and all products of separable metric spaces. For basic results of the class Oz see (1 and 2). Blair (1) asked if the spaces sR, sQ and sQ — Q are in Oz. In (6) Terada characterizes a class of spaces whose Stone-Cech compactifications are in Oz. As an application of his characterizations he showed that both sR and sQ do not belong to Oz. E. K. van Douwen (4) has proved that sQ — Q does not belong to Oz. In this note we shall prove that for Y dense in R, the spaces sY and sY — Y