Lexicographic products on linearly ordered topological spaces

Abstract

A linear order < defined on a set X is a relation such that: • For all x EX, x < x does not hold. (Non-reflexivity) • x < y and y < z --} x < z (Transitivity) • For 011 x, y C X, we have x < y or y < x or x < y (Totality) The linear order topology defined on X is the topology generated by the family of all open intervals in X, and the obtained topological space is called a linearly ordered topological space (LOTS). A subspace of a LOTS is called a generalized ordered space (or GO-space). Clearly, a LOTS is also a GO-space. In this thesis, we study several well-known properties like compactness, paracompactness, etc in GO-spaces. In addition, we will study lexicographic products of LOTS. In Chapter 1, we provide some utility results for the study of properties of GO-spaces. The most important results are those about gaps and pseudo gaps in GO-spaces. As we will see, in the following chapters, gaps and pseudo gaps are intensively used to characterize different properties of GO-spaces. In addition, gaps also play a critical role in our study of lexicographic products of LOTS. Since gaps can be considered to be points in the Dedekind compactification of a LOTS or GO-space, our study about gaps are mainly in the context of Dedekind compactification. In Theorem 1.5.5, we show that a gap in the lexicographic product must be generated from a gap in one of its factor spaces. This result is quite intuitive, but the proof is not so simple. Then in Lemma 1 5 7 and Lemma 1 5 10, we discuss circumstances, in which a gap in a factor space will give a gap in the lexicographic product.