THEORY OF GENERALIZED CONNECTEDNESS (g-Tg-CONNECTEDNESS) IN GENERALIZED TOPOLOGICAL SPACES (Tg-SPACES)

Abstract

In this paper, the definitions of novel classes of generalized connected sets (briefly, g-Tg-connected sets) and generalized disconnected sets (briefly, g-Tg-disconnected sets) in generalized topological spaces (briefly, Tg-spaces) are defined in terms of generalized sets (briefly, g-Tg -sets) and, their properties and characterizations with respect to set-theoretic relations are presented. The basic properties and characterizations of the notions of local, pathwise, local pathwise and simple g-Tg-connectedness are also presented. The study shows that local pathwise g-Tg -connectedness implies local g-Tg-connectedness, pathwise g-Tg-connectedness implies g-Tg-connectedness, and g-Tg-connectedness is a Tg-property. Diagrams establish the various relationships amongst these types of g-Tg-connectedness presented here and in the literature, and a nice application supports the overall theory.

Keywords

• Generalized topological space (Tg-spaces)
• generalized local connectedness (local g-Tg-connectedness)
• generalized pathwise connectedness (pathwise g-Tg-connectedness)
• generalized local pathwise connectedness (local pathwise g-Tg-connectedness)
• generalized simple connectedness (simple g-Tg-connectedness)
• generalized components (g-Tg-components)

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