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Some New Variants of Relative Regularity Via Regularly Closed Sets Hindawi Journal of Mathematics Volume 2021, Article ID 7726577, 6 pages https://doi.org/10.1155/2021/7726577 Research Article SomeNewVariantsofRelativeRegularityviaRegularlyClosedSets Sehar Shakeel Raina and A. K. Das School of Mathematics, Shri Mata Vaishno Devi University, Katra 182320, Jammu and Kashmir, India Correspondence should be addressed to A. K. Das; [email protected] Received 20 April 2021; Accepted 6 May 2021; Published 25 May 2021 Academic Editor: Fr´ed´eric Mynard Copyright © 2021 Sehar Shakeel Raina and A. K. Das. (is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Every topological property can be associated with its relative version in such a way that when smaller space coincides with larger space, then this relative property coincides with the absolute one. (is notion of relative topological properties was introduced by Arhangel’skii and Ganedi in 1989. Singal and Arya introduced the concepts of almost regular spaces in 1969 and almost completely regular spaces in 1970. In this paper, we have studied various relative versions of almost regularity, complete regularity, and almost complete regularity. We investigated some of their properties and established relationships of these spaces with each other and with the existing relative properties. 1. Introduction important versions of regularity studied by different to- pologists in general topology which need to be studied in Arhangel’skii and Ganedi in [1] introduced the concept that relative sense. Complete regularity is an important classical with each topological property P one can associate a relative version of regularity in which a point and a closed set can be version of it formulated in terms of location of Y in X in such functionally separated. Almost regularity and almost com- a natural way that when Y coincides with X, then this plete regularity are some other versions of regularity in- relative property coincides with P. (e motivation behind troduced by Singal and Arya in [11, 12]. In this paper, we this study is to investiagte how the smaller space is located in have studied these concepts in relative sense. We have also the larger one. In past few years, a lot of work has been done defined relative almost strongly regular and relative almost on relative topology. Grothendieck [2], Tkachuk [3], Chi- superregular spaces. Relationships of these notions with each gogidze [4], Ranchin [5], Dow and Varmeer [6], Das and other and with the existing notions have also been studied Bhat [7], Das and Raina [8], Just and Tartir [9], and so on are and counterexamples are provided wherever needed. It is few names who remarkably contributed in this area. An observed that relative almost regularity is the most gener- important feature of relative topology is that a whole pack of alized among all the versions of relative regularity. relative topological properties can be relativised from a single absolute topological property. It depends on the 2. Basic Definitions and Preliminaries absolute property that how many interesting versions of relative properties can be created out of it. Let X be a topological space and let A ⊂ X. A set U ⊂ X is o In general topology, regularity and normality play a very said to be regularly open [13] if U � U . (e complement of prominent role in studying topological properties which a regularly open set is called regularly closed. A topological were widely studied by researchers. Like normality, relative space X is said to be almost normal [11] if for every pair of version of regularity has also been studied by Arhangel’skii disjoint sets A and B one of which is regularly closed and in [10]. In a survey article, Arhangel’skii also studied some other is closed, there exist disjoint open sets U and V such versions of relative regularity like relative strong regular, that A ⊂ U and B ⊂ V. A topological space X is said to be relative superregular, and so on. (ere are still many completely regular if for every closed set A of X and every 2 Journal of Mathematics point x ∈ X such that x ∉ A, there exists a continuous predecessor of p also belong to U. Let function f on X into the closed interval [0; 1] such that Y � f2; 6; 10; 14; ... :; 2(2n + 1); ...g, where n � 0; 1; 2; ...., Y f(x) � 0 and f(A) � 1. A topological space X is said to be is not almost strongly regular in X becausef 2; 3; 4 g is reg- almost regular [12] if for every regularly closed set A and a ularly closed in X, 1 ∈ X, and there do not exist disjoint open point x ∉ A, there exist disjoint open sets U and V such that sets in X separatingf 2; 3; 4 g ∩ Y � f2 g and 1. But Y is almost x ∈ U and A ⊂ V. A topological space X is said to be almost regular in X. completely regular [11] if for every regularly closed set A of X and a point x ∈ X such that x ∉ A, there exists a continuous Example 3. Let X � fa; b; c g and T � �fa g;f b g;f a; b g; X; ϕ� be function f on X into the closed interval [0; 1] such that a topology on X. Let Y � fc g. X is not almost normal and Y is f(x) � 0 and f(A) � 1. not almost strongly regular in X because fb; c g is regularly Let Y ⊂ X. Y is said to be T1 in X or relative T1 [10] if for closed in X and for fb; c g ∩ Y � fc g and a ∈ X, there do not every y ∈ Y, �y � is closed in X. Y is said to be Hausdorff in X exist disjoint open sets in X separating them. But Y is or relative Hausdorff [10] if for every pair of distinct points in vacuously almost superregular in X as fa; c g;f b; c g are the Y there exist disjoint open sets separating them. Y is said to only regularly closed sets in X and there does not exist any be regular in X [10] or relative regular if for every closed set y ∈ Y which does not belong to these sets. A of X and a point y ∈ Y such that y ∉ A there exist disjoint Arhangel’skii in [10] defined relative superregular and open subsets U and V of X such that A ∩ Y ⊂ U and y ∈ V. Y relative strongly regular as follows: Y ⊂ X is said to be is said to be normal in X or relative normal [10] if for every superregular in X if for every closed set A in X and a point pair of disjoint closed sets A and B of X, there exist disjoint y ∈ Y such that y ∉ A, there exist disjoint open sets U and V open subsets U and V of X such that A ∩ Y ⊂ U and in X such that A ⊂ U and y ∈ V. And Y is said to be strongly B ∩ Y ⊂ V. regular in X if for every closed set A in X and a point x ∈ X such that x ∉ A, there exist disjoint open sets U and V in X 3. Relative Almost Regular Spaces such that A ∩ Y ⊂ U and x ∈ V. (e idea of defining relative almost superregular and relative almost strongly regular is Definition 1. Let X be a topological space and Y ⊂ X. Y is taken from these concepts. It is clear from the definitions said to be almost regular in X or relative almost regular that relative strongly regular implies relative almost strongly (almost superregular in X or relative almost superregular) if regular and relative superregular implies relative almost for every regularly closed set A in X and a point y ∈ Y such superregular. But from the following example, it can be that y ∉ A, there exist disjoint open sets U and V in X such easily checked that none can be reversed. that A ∩ Y ⊂ U and y ∈ V (A ⊂ U and y ∈ V). Example 4. Let X � fa; b; c g and τ � �fb g;f a; b g;f b; c g; X; ϕ� Definition 2. Let X be a topological space and Y ⊂ X. Y is be a topology on X. Let Y � fb; c g. Since there does not exist said to be almost strongly regular in X or relative almost any regularly closed set in X, Y is vacuously relative almost strongly regular if for every regularly closed set A in X and a superregular and relative almost strongly regular in X. But Y point x ∈ X such that x ∉ A, there exist disjoint open sets U is neither relative superregular nor relative strongly regular and V in X such that A ∩ Y ⊂ U and x ∈ V. because for the closed set fc g in X and b ∈ X, there do not Clearly, if X is almost regular, then Y is almost regular in exist disjoint open sets in X satisfying the conditions of X. If Y is regular in X, then Y is almost regular in X. If Y is relative superregularity or relative strong regularity. almost superregular in X (almost strongly regular in X), then Y is almost regular in X. But none of these implications Theorem 1. Let Y ⊂ X. .en, the following statements are can be reversed. Also, Y can be almost superregular in X equivalent. without being almost strongly regular in X, and vice versa, Y can be almost strongly regular in X without being almost (a) Y is relative almost superregular.
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