Generalized b Compactness and Generalized b Connectedness in Topological Spaces

Abstract

This paper deals with that * -gb compact spaces and their properties are studied. The notion of connectedness in topological spaces is also introduced and their properties are studied.

A New Notion of Hexa Regular Closed Sets via HexaTopological Space

“…The single topology is extended to bi-topological space, tri-topological space, quad-topological space, penta-topological space [4,5,6,7 ], and hexa -topological space (a space with six topologies τ 1 , τ 2 , τ 3 , τ 4 , τ 5 , τ 6 , and the bi-topological idea was first presented by Kelly [4], Tri-topological space was first started by Kovar [5], Quad topological space was researched by Mukundan [6] and Penta topological space by G. A. Khan [7] and hexa -topological space by Chandra and Pushpalatha [8] introduced and researched the idea of h -open sets in h -topological spaces, h -continuous. w h -closed sets introduced by G.Sindhu and T.Kalaiselvi [9] i=1 τ i and its complement is said to be hexa -closed set( h-closed for short). The set M together with h-Topology T h is called hexa topological space and is denoted by (M, T h ) where T h = (τ 1 , τ 2 , τ 3 , τ 4 , τ 5 , τ 6 ) Definition 1.2 If (M, T h ) is a hexa -topological space and A ∈ M Then 1.…”

gr-connectedness in Topological Spaces

“…Data structure is important in the applications on the computer which relates real topological situation. Connectedness and compactness were studied by Whyburn G. T [2], considering the assumption of Hausdorff.The basic properties of connectedness had been discussed through many researchers [3][4][5][6][7][8][9][10][11][12]. Mathematicians are motivated through the notions of connectedness to extend the existing concepts.…”

Theory of g-Tg-Compactness

“…Among the most fundamental topological properties (briefly, T -properties relative to ordinary topology, and T g -properties relative to generalized topology), the T -properties 1 , termed T-compactness and g-T-compactness in T -spaces (ordinary and generalized compactness in ordinary topological spaces) and the T g -properties termed T g -compactness and g-T g -compactness in T g -spaces (ordinary and generalized compactness in generalized topological spaces) are verily the most important invariant properties [3,4,5,7,15,16,17,20,21,22,23,25,28,29,30,31,32,33,34,35,36]. In actual truth, T-compactness is an absolute property of a T-set [2,13,38,33], and g-T-compactness, T g -compactness and g-T g -compactness, respectively, are absolute properties of a g-T-set, a T g -set, and a g-T g -set [18,25,29].…”