Theory of Generalized Separation Axioms in Generalized Topological Spaces

Abstract

In this paper, a new class of generalized separation axioms (briefly, g-Tg-separation axioms) whose elements are called g-T g,K , g-T g,F , g-T g,H , g-T g,R , g-T g,N -axioms is defined in terms of generalized sets (briefly, g-Tg-sets) in generalized topological spaces (briefly, Tg-spaces) and the properties and characterizations of a Tg-space endowed with each such g-T g,K , g-T g,F , g-T g,H , g-T g,R , g-T g,N -axioms are discussed. The study shows that g-T g,F -axiom implies g-T g,K -axiom, g-T g,H -axiom implies g-T g,F -axiom, g-T g,R -axiom implies g-T g,H -axiom, and g-T g,N -axiom implies g-T g,R -axiom. Considering the T g,K , T g,F , T g,H , T g,R , T g,N -axioms as their analogues but defined in terms of corresponding elements belonging to the class of open, closed, semi-open, semiclosed, preopen, preclosed, semi-preopen, and semi-preclosed sets, the study also shows that the statement Tg,α-axiom implies g-T g,α -axiom holds for each α ∈ {K, F, H, R, N}. Diagrams expose the various implications amongst the classes presented here and in the literature, and a nice application supports the overall theory.

Theory of Generalized Compactness in Generalized Topological Spaces: Part II. Countable, Sequential and Local Properties

“…Standard references for notations and concepts are [9][10][11][12]. The mathematical structures T def = (Ω, T ) and T g def = (Ω, T g ) , respectively, are T , T g -spaces [9], on both of which no separation axioms are assumed unless otherwise mentioned [4,10]. A T g -space T g = (Ω, T g ) endowed with a g-T g,H -axiom is called a g-T (H) g -space g-T (H)…”

“…) [9][10][11]. The sets I 0 n , I * n and I 0 ∞ , I * ∞ , respectively, are finite and infinite index sets [9].…”

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

“…called T g -connectedness and g-T g -connectedness in T g -spaces (ordinary and generalized connectedness in generalized topological spaces) are no doubt the most important invariant properties [1,2,3]. Indeed, T-connectedness is an absolute property of a T-set [1,4,5], and g-T-connectedness, T g -connectedness and g-T gconnectedness, respectively, are absolute properties of a g-T-set, a T g -set, and a g-T g -set [3,6,7,8,9,10,11]. Typical examples of g-T-connectedness in T -spaces are α, β, γ-connectedness [12,13,14]; examples of T g -connectedness in T g -spaces are semi * α, s, gb-connectedness [2,15,16], whereas examples of g-T g -connectedness in T g -spaces are bT µ , µ-rgb, π p-connectedness [17,18,19], among others.…”

“…In view of the above references, it would appear that, from every new type of g-T g -set introduced in a T g -space, there can be introduced a new type of g-T gconnectedness in the T g -space. Having introduced a new class of g-T g -sets and studied from it some T g -properties in a T g -space [6,7,8,9,10], it seems, therefore, reasonable to introduce a new type of g-T g -connectedness in the T g -space and discuss its T g -properties. In this paper, we attempt to make a contribution to such a development by introducing a new theory, called Theory of g-T g -Connectedness, in which it is presented a new generalized version of T g -connectedness in terms of the notion of g-T-set, discussing the fundamental properties and giving its characterizations on this ground.…”

“…Preliminaries. Notations and notions not presented below are found in the standard references [6,7,8,9,10]. Everywhere, T , T g -spaces are designated by the topological structures T def = (Ω, T ) and T g def = (Ω, T g ), respectively, on both of which no separation axioms are assumed unless otherwise mentioned [8,10,26,27,28].…”