Salleh, Zabidin
(2008)
Generalizations of Lindelöf Properties in Bitopological Spaces.
PhD thesis, Universiti Putra Malaysia.
Abstract
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 pseudometrics, 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 regularLindelöfness in bitopological
spaces motivated by the known ideas of Lindelöfness, generalized Lindelöfness and
generalized regularLindelöfness in topological spaces.
There are four kinds of pairwise Lindelöf space namely Lindelöf, BLindelöf, s
Lindelöf and pLindelöf spaces that depend on open, iopen, τ
1
τ2
open and popen
covers respectively introduced by Reilly in 1973, and Fora and Hdeib in 1983. For
instance, a bitopological space X is said to be pLindelöf if every popen 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 regularLindelöfness to bitopological
spaces. This leads to the classes of pairwise nearly regularLindelöf, pairwise almost
regularLindelöf and pairwise weakly regularLindelöf 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 δcontinuity 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 regularLindelöf
and pairwise weakly regularLindelöf properties; and pairwise Rmaps preserve the
pairwise nearly regularLindelöf 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 Pspace property;
under which these generalized pairwise covering properties become finitely
productive.
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