Generalizations of ѵ-Lindelöf generalized topological spaces

A significant contribution to the theory of generalized open sets was made by Császár (1997), he introduced the concept of generalized neighborhood systems and generalized topological spaces. Further, he showed that the fundamental definitions and the major part of numerous statements and structures in the set topology can be formulated by replacing topology with the generalized topology. During this work, we introduce two kinds of v-separations axioms in generalized topological spaces, which are generated by v-regular open sets; namely, almost G-regular and G-semiregular. Therefore, properties and characterization are introduced. Relation among theses generalized topological spaces and some other v-separations axioms are considered. These two kinds of v-separations axioms are essential to relies some results in our work. We define three types of v-Lindelöf generalized topological spaces. Namely; nearly v- Lindelöf, almost v-Lindelöf and weakly v-Lindelöf (briefly. nv-Lindelöf, αv-Lindelöf and wv-Lindelöf). Some properties and characterizations of these three generalizations of v-Lindelöf generalized topological spaces are given. The relations among them are studied and some counterexamples are shown in order to prove that the studies of generalizations are proper generalizations of v-Lindelöf generalized topological spaces. Subspaces and subsets of these generalized topological spaces are studied. We show that some subsets of these generalized topological spaces inherit these covering properties and some others they do not. Moreover, G-semiregular property on these spaces is studied to establish that all of these properties are G-semiregular properties on the contrary of v-Lindelöf property which is not a G-semiregular property.Mappings and generalized continuous functions are also studied on these generalizations and we prove that these properties are generalized topological properties. Relations and some properties of many decompositions of generalized continuity that recently defined and studied are given. Counterexamples are also given to establish the relations among these generalizations of generalized continuity. We show that some proper mappings preserve these generalized topological properties such as (δ;δ)- continuity preserves nv-Lindelöf property. θ(v;μ)-continuity preserves nv-Lindelöf property. Almost (v;μ)-continuity preserves wv-Lindelöf property. Moreover, we give some conditions on the functions or on the generalized topological spaces to prove that weak forms of generalized continuity preserve some of these covering properties under these conditions. The product property on these generalizations is also studied. We show that these topological properties are not preserved by product, even under a finite product. Some conditions are given on these generalizations to prove that these properties are preserved by finite product under these conditions. We show that, in weak P--G-spaces, finite product of nv-Lindelöf generalized topological spaces is nv-Lindelöf and finite product of wv-Lindelöf generalized topological spaces is αv-Lindelöf. Using the notions of generalized topology and hereditary classes, in order to we define some of generalizations of vh -Lindelöf, namely; nvH -Lindelöf, αvH -Lindelöf and wvH -Lindelöf hereditary generalized topological spaces. Moreover, we investigate basic properties of the concepts, the relation among them, their relation to known concepts and their preservation by functions properties. Soft generalized topological spaces played an important role in recently years. Some basic definitions and important results related to soft generalized topology on an initial soft set are given, the concept of soft n-Lindelöf soft generalized topological spaces is introduced. Basic properties and relation between n-Lindelöf spaces in generalized topological spaces and soft n-Lindelöf soft generalized topological spaces are showed. We can say that a soft n-Lindelöf soft generalized topological spaces gives a parametrized family of n-Lindelöf generalized topological spaces on the initial universe.

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