Generalizations of Lindelöf Properties in Bitopological Spaces - Salleh, Zabidin (2008)
A bitopological space (X, τ 1, τ 2) is a set X together with two (arbitrary) topologies τ 1 and τ 2 defined on X. The first significant investigation into bitopological spaces was launched by J. C. Kelly in 1963. He recognized that by relaxing the symmetry condition on pseudo-metrics, two topologies were induced by the resulting quasipseudo- metrics. Furthermore, Kelly extended some of the standard results of separation axioms in a topological space to a bitopological space. Some such extension are pairwise regular, pairwise Hausdorff and pairwise normal spaces. There are several works dedicated to the investigation of bitopologies; most of them deal with the theory itself but very few with applications. In this thesis, we are concerned with the ideas of pairwise Lindelöfness, generalizations of pairwise Lindelöfness and generalizations of pairwise regular-Lindelöfness in bitopological spaces motivated by the known ideas of Lindelöfness, generalized Lindelöfness and generalized regular-Lindelöfness in topological spaces. There are four kinds of pairwise Lindelöf space namely Lindelöf, B-Lindelöf, s- Lindelöf and p-Lindelöf spaces that depend on open, i-open, τ 1 τ2 -open and p-open covers respectively introduced by Reilly in 1973, and Fora and Hdeib in 1983. For instance, a bitopological space X is said to be p-Lindelöf if every p-open cover of X has a countable subcover. There are three kinds of generalized pairwise Lindelöf space namely pairwise nearly Lindelöf, pairwise almost Lindelöf and pairwise weakly Lindelöf spaces that depend on open covers and pairwise regular open covers. Another idea is to generalize pairwise regular-Lindelöfness to bitopological spaces. This leads to the classes of pairwise nearly regular-Lindelöf, pairwise almost regular-Lindelöf and pairwise weakly regular-Lindelöf spaces that depend on pairwise regular covers. Some characterizations of these generalized Lindelöf bitopological spaces are given. The relations among them are studied and some counterexamples are given in order to prove that the generalizations studied are proper generalizations of Lindelöf bitopological spaces. Subspaces and subsets of these spaces are also studied, and some of their characterizations investigated. We show that some subsets of these spaces inherit these generalized pairwise covering properties and some others, do not. Mappings and generalized pairwise continuity are also studied in relation to these generalized pairwise covering properties and we prove that these properties are bitopological properties. Some decompositions of pairwise continuity are defined and their properties are studied. Several counterexamples are also given to establish the relations among these generalized pairwise continuities. The effect of mappings, some decompositions of pairwise continuity and some generalized pairwise openness mappings on these generalized pairwise covering properties are investigated. We show that some proper mappings preserve these pairwise covering properties such as: pairwise δ-continuity preserves the pairwise nearly Lindelöf property; pairwise θ- continuity preserves the pairwise almost Lindelöf property; pairwise almost continuity preserves the pairwise weakly Lindelöf, pairwise almost regular-Lindelöf and pairwise weakly regular-Lindelöf properties; and pairwise R-maps preserve the pairwise nearly regular-Lindelöf property. Moreover, we give some conditions on the maps or on the spaces which ensure that weak forms of pairwise continuity preserve some of these generalized pairwise covering properties. Furthermore, it is shown that all the generalized pairwise covering properties are satisfy the pairwise semiregular invariant properties where some of them satisfy the pairwise semiregular properties. On the other hand, none of the pairwise Lindelöf properties are pairwise semiregular properties. The productivity of these generalized pairwise covering properties are also studied. It is well known by Tychonoff Product Theorem that compactness and pairwise compactness are preserved under products. We show by means of counterexamples that in general the pairwise Lindelöf, pairwise nearly Lindelöf and similar properties are not even preserved under finite products. We give some necessary conditions, for example the P-space property; under which these generalized pairwise covering properties become finitely productive.