ORDER PROPERTIES OF PRODUCT SPACES AND QUOTIENT SPACES OF ORDERED TOPOLOGICAL LINEAR SPACES

Abstract

Very few we find in literature the order properties that product spaces of ordered linear spaces and ordered normed spaces carry from their component spaces. Some authors have studied certain order properties of ordered linear spaces which are carried by their quotient spaces. A little work is done in connection with the order properties of ordered normed spaces that the quotient spaces inherit from their parent spaces. In this thesis we have studied several order properties of ordered linear spaces and ordered normed spaces that are preserved with or without conditions in the formation of products. We have also studied several order properties of quotient spaces of ordered linear spaces and ordered normed spaces by discussing inheritance of order properties of the parent space X to its quotient space X/E. This thesis consists of six chapters. Chapter 1 includes some basic definitions and results. Chapter 2 provides almost all possible results available in the literature regarding the topological properties that are preserved in the formation of products. Chapter 3 is about the quotient spaces and includes several results available in the literature for the quotient spaces of (i) topological spaces (ii) linear spaces (iii) topological groups and (iv) topological linear spaces. Chapter 4 and Chapter 5 include the research work. Six research papers have been published from the work contained in these chapters; out of which four are published in the journals approved by the Higher Education Commission (HEC). Two more research papers are being submitted in some journals of international repute. Chapter six also includes the research work and presents only results from a past publication.