Theory of Generalized Compactness in Generalized Topological Spaces: Part I. Basic Properties

Abstract

In this paper, a novel class of generalized compact sets (briefly, g-Tg-compact sets ) in generalized topological spaces (briefly, 
 Tg-spaces ) is studied. The study reveals that g-Tg -compactness implies ordinary compactness (briefly, Tg -compactness ) in 
 Tg-spaces, and such statement implies its analogue in ordinary topological spaces (briefly, T -spaces ). Diagrams establish the various relationships amongst these types of g-Tg -compactness presented here and in relation to other types of g-T -compactness
 in T-spaces presented in the literature of Tg -spaces, 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].…”

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

“…respectively, are simply said to be a g-T g -covering, a g-T g -open covering, and a g-T g -closed [9,11]. The map…”

“…S g,ϑ(α) [9,11]. The T g -set S g ⊂ T g of a T g -space T g is g-ν-T g -compact if and only if, for every…”

“…Definition 2.8 ( g-T g -Accumulation Point [9,11]) A point ξ ∈ T g of a T g -space T g = (Ω, T g ) is called a " g-T g -accumulation point" (or " g-T g -limit point", " g-T g -cluster point", " g-T gderived point") of a T g -set S g ⊂ T g of T g if and only if every…”