On Е-(δ-ß)-Connectedness and Strongly Е-(δ-ß)-Connectedness Via Е-(δ-ß)-Open Sets and Their Applications

Abstract

Connectedness is now one of the most important, useful, and essential notions not only in general topology but also in other branches of advanced mathematics. In our thesis, study new concepts of separated sets called, Е-separated and δ-ß-separated sets via another generalized open sets namely, Е-open and δ-ß-open sets respectively, as well with these concepts display new classes of connected spaces which is called, strongly Е-connected and strongly δ-ß-connected spaces. Also, we introduce several characterizations concerning of such classes of connected spaces with some Е-separation and δ-ß-separation axioms with compact spaces. In addition discuss the behavior of Е-connected and δ-ß-connected spaces with respect to several types of well-known mappings and construct new topological spaces on a connected graph. Furthermore, new types of connectedness namely, Ec-connectedness and δ-ßc-connectedness which are stronger forms of E-connectedness (resp. δ-ß-connectedness) and weaker forms of connectedness are presnted. The relationships between these kinds of connectedness and other different weaker and stronger forms of connectedness are discussed.