nv-Lindelöfness

Our work aims to study nearly ν-Lindelöf (briefly. nν-Lindelöf) space in generalized topological spaces. Moreover, some mappings and decompositions of continuity are studied. The main result that we obtained on is the effect of (δ,δ’)-continuous function on nν-Lindelöf space

Cardinalities of some Lindelöf and ω1-Lindelöf T1/T2-spaces

AbstractWe show that every first-countable countably paracompact Lindelöf T1-space has cardinality at most c; every first-countable ω1-Lindelöf Hausdorff space has cardinality at most 2c; every realcompact first-countable ω1-Lindelöf space has cardinality at most c. In all these results, first countability can be replaced by countable tightness plus either countable or countable closed pseudocharacter. We also show that the Lindelöf number of every ω1-Lindelöf regular space of countable tightness is at most c

Mappings on weakly Lindelöf and weakly regular-Lindel¨of spaces

[EN] In this paper we study the effect of mappings and some decompositions of continuity on weakly Lindelöf spaces and weakly regular-Lindelöf spaces. We show that some mappings preserve these topological properties. We also show that the image of a weakly Lindelöf space (resp. weakly regular-Lindelöf space) under an almost continuous mapping is weakly Lindelöf (resp. weakly regular-Lindelöf). Moreover, the image of a weakly regular-Lindelöf space under a precontinuous and contracontinuousmapping is Lindelöf.Fawakhreh, AJ.; Kiliçman, A. (2011). Mappings on weakly Lindelöf and weakly regular-Lindel¨of spaces. Applied General Topology. 12(2):135-141. doi:10.4995/agt.2011.1647.SWORD13514112

Mappings and decompositions of continuity on almost Lindelöf spaces

A topological space X is said to be almost Lindelöf if for every open cover {Uα:α∈Δ} of X there exists a countable subset {αn:n∈ℕ}⊆Δ such that X=∪n∈ℕCl(Uαn). In this paper we study the effect of mappings and some decompositions of continuity on almost Lindelöf spaces. The main result is that a image of an almost Lindelöf space is almost Lindelöf

Free sequences and the tightness of pseudoradial spaces

Let F(X) be the supremum of cardinalities of free sequences in X. We prove that the radial character of every Lindelöf Hausdorff almost radial space X and the set-tightness of every Lindelöf Hausdorff space are always bounded above by F(X). We then improve a result of Dow, Juhász, Soukup, Szentmiklóssy and Weiss by proving that if X is a Lindelöf Hausdorff space, and Xδ denotes the Gδ topology on X then t(Xδ) ≤ 2 t ( X ). Finally, we exploit this to prove that if X is a Lindelöf Hausdorff pseudoradial space then F(Xδ) ≤ 2 F ( X )

Generalizations of Lindelöf Properties in Bitopological Spaces

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

Locally compact linearly Lindelöf spaces

summary:There is a locally compact Hausdorff space which is linearly Lindelöf and not Lindelöf. This answers a question of Arhangel'skii and Buzyakova

On pairwise nearly Lindelof bitopological spaces

We shall introduce and study the pairwise nearly Lindelöf bitopological spaces and obtain some results. Moreover, we study the pairwise nearly Lindelöf subspaces and subsets and investigate some of their characterizations. We also show that the pairwise nearly Lindelöf space is not a hereditary property. Some examples will be considered in order to establish some of their relationships. Finally, certain conditions on which a bitopology reduced to a single topology are investigated