Dense Sδ-diagonals and linearly ordered extensions

Abstract

Abstract. The notion of the S δ -diagonal was introduced by H. R.Bennett to study the quasi-developability of linearly ordered spaces. In an earlier paper, we obtained a characterization of topological spaces with an S δ -diagonal and we showed that the S δ -diagonal property is stronger than the quasi-G δ -diagonal property. In this paper, we define a dense S δ -diagonal of a space and show that two linearly ordered extensions of a generalized ordered space X have dense S δ -diagonals if the sets of right and left looking points are countable.Keywords: S δ -diagonal, dense S δ -diagonal, linearly ordered space (LOTS), generalized ordered space (GO-space), linearly ordered extension.2000 AMS Classification: 54F05. S δ -diagonalsWe review in this section the definitions of S δ -set and S δ -diagonal, and state our results obtained in [5].The following definition is a generalization of a G δ -set and was introduced by H. R. Bennett [2] to study the quasi-developability of linearly ordered (topological) spaces. Definition 1.1. Let X be a topological space. A subset A of X is called an S δ -set if there exists a countable collection {U (1), U (2), . . .} of open subsets of X such that, for two points p ∈ A and q ∈ X \ A, there exists an n such that p ∈ U (n) and q / ∈ U (n).It is easy to see that a G δ -set is an S δ -set. Hence the notion of S δ -set is a generalization of G δ -set. See [3] for a description of S-normal spaces whose closed subsets are S δ -sets.