On Binary Generalized Topological Spaces

Abstract

Abstract. In this paper, we introduce and study the new concept of binary generalized topological spaces. Also we examine some binary generalized topological properties. Furthermore, we define and study some forms of binary generalized continuous functions and we investigate the relationships between these functions and their relationships with some other functions.

Binary Regular ^ Generalized Closed Sets In Binary Topological Spaces

“…Definition 2.1 [2]: Let X and Y be any two non empty sets. A binary generalized topology from X to Y is a binary structure µb  P(X)  P(Y) that satisfies the following axioms:…”

“…Then (i) (A, B)  (C, D) if A  C and B  D. (ii) (A, B)  (C, D) =(A  C, B  D). (iii) (A, B)  (C, D) =(A  C, B  D).Definition 2.4[2]: Let (X, Y, µb) be a binary generalized topological space and (x, y)  X  Y, then a subset (A, B) of (X, Y) is called a binary generalized neighbourhood of (x, y) if there exists a binary generalized open set (U, V) such that (x, y)  (U, V)  (A, B).Definition 2.5[2]: Let (X, Y, µb) be a binary generalized topological space and A  X, B  Y. Let (A, B) 1* = {Aα :(Aα, Bα)is binary generalized closed and (A, B)  (Aα, Bα)} and (A, B) 2* = {Bα :(Aα, Bα)is binary generalized closed and (A, B)  (Aα, Bα)}.…”

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].…”

h-CONVEXITY IN METRIC LINEAR SPACES

“…These types of convexities were studied in [2] and [4]. It is known that a uni-formly convex metric linear space is locally uniformly convex and a locally uniformly convex metric linear space is strictly convex.…”