Some Applications of Supra Preopen Sets

Abstract

The aim of this work is to define some concepts on supra topological spaces using supra preopen sets and investigate main properties. We started this paper by correcting some results obtained in previous study and presenting further properties of supra preopen sets. Then, we introduce a concept of supra prehomeomorphism maps and discuss its main properties. After that we explore the concepts of supra limit and supra boundary points of a set with respect to supra preopen sets and examine their behaviours on the spaces that possess the difference property. Finally, we formulate the concepts of supra pre-Ti-spaces i=0,1,2,3,4 and give completely descriptions for each one of them. In general, we study their main properties in detail and show the implications of these separation axioms among themselves as well as with STi-space with the help of some interesting examples.

Limit Points and Separation Axioms with Respect to Supra Semi-open Sets

“…Therefore there exist two supra semi-open subsets U and V of (X, µ) containing x 1 and x 2 , respectively. According to Definition (17), U × Y and V × Y are two supra open subsets of (X × Y, C) containing (x 1 , y 1 ) and (x 2 , y 2 ) such that (x 1 , y 1 ) ∈ V × Y and (x 2 , y 2 ) ∈ U × Y . Hence, (X × Y, C) is ST 1 .…”

“…In other words, their definitions were formulated using supra interior and supra closure operators instead of interior and closure operators. These generalizations have been utilized to define new versions of compactness, connectedness and separation axioms, see, for example [5,6,17,22,25]. The class of supra R-open sets has been studied in [4,11] under the name of somewhere dense sets.…”

Applications of some operators on supra topological spaces

“…Al-shami [23] introduced the concept of paracompactness on STSs and explored main properties. Recently, the authors of [24][25][26][27] have employed some generalizations of supra open sets given in [28][29][30][31] to study limit points and separation axioms on STSs. They have provided various interesting examples to explain the given relationships and results.…”

Fuzzy Ideal Supra Topological Spaces

“…en, Al-Shami [9,10] and Al-shami et al [11] presented new types of supra compact spaces using supra α-open, supra semiopen, and supra preopensets. Later, the authors of [12][13][14][15] employed some generalizations of supra opensets to investigate several kinds of supra limit points of a set and supra T i spaces (i � 0, 1, 2, 3, 4). In fact, they provided many interesting examples to show the validity of the obtained results.…”