A note on some generalized closure and interior operators in a topological space

Abstract

Abstract. If X is a topological space and A ⊆ X, then the number of distinct sets that can be obtained from A by using all possible compositions for operators iγ , cγ (where γ = σ, π, α, β) introduced by Császár is at the most 25. Explicit expressions for these sets are provided. An example is provided where all the 25 different sets are determined. The result is also discussed for special cases such as when the space is extremally disconnected, resolvable, open-unresolvable, and partition spaces.

Theory of Generalized Sets in Generalized Topological Spaces

“…Several other specific classes of g-T, g-T g -sets have been defined and investigated by other authors for various purposes from time to time in the literature of T , T g -spaces [9,[22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]. The fruitfulness of all these references have made significant contributions to the theory of T , T g -spaces, among others.…”

Theory of g-Tg-Interior and g-Tg-Closure Operators: Definitions, Essential Properties, and Commutativity

“…Just as the concepts of T, g-T-interior 1 operators in T -spaces (ordinary and generalized interior operators in ordinary topological spaces) and T, g-T-closure operators in T -spaces (ordinary and generalized closure operators in ordinary topological spaces) are essential operators in the study of T-sets in T -spaces (arbitrary sets in ordinary topological spaces) [CJK04,Cs6,Cs5,Cs8,Cs7,GS17,JN19,Kal13,Lev70,Lev63,Lev61,MG16], so are the concepts of T g , g-T g -interior operators in T g -spaces (ordinary and generalized interior operators in generalized topological spaces) and T g , g-T g -closure operators in T g -spaces (ordinary and generalized closure operators in generalized topological spaces) essential operators in the study of T g -sets in T g -spaces (arbitrary sets in generalized topological spaces) [DB11,GS14,Min10,Min05,Mus17].…”