EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 14, No. 4, 2021, 1275-1282 ISSN 1307-5543 – ejpam.com Published by New York Business Global On ψgs-closed Sets in Bitopological Spaces Lezel Mernilo Tutanes Mathematics Department, College of Arts and Sciences, Bukidnon State University, Malaybalay City, Bukidnon, Philippines Abstract. In this paper, the properties of ψgs-closed sets in bitopological spaces are investigated. The relationships between ψgs-closed set and other closed sets in bitopological spaces are estab- lished and some properties of ψgs-closure and ψgs-interior are provided. 2020 Mathematics Subject Classifications: 18F60, 05C69, 30H80 Key Words and Phrases: Bitopological Spaces, ψgs-closed sets, ψgs-interior, ψgs-closure 1. Introduction Over the years, many researchers have introduced different types of sets in topological spaces. One of these sets is the semi-open set, which was introduced and studied by Levine [12]. Thereafter, the notion of generalized closed sets (briefly, g-closed set) in topological spaces was introduced and investigated in [13]. In 2000, the concepts between closed sets and g-closed sets in topological spaces were studied in [10] and a few years later, the same author [11] studied ψ- closed sets in topological spaces. Ramya and Parvathi introduced a new concept of ψ- generalized closed (briefly, ψg-closed) sets in topological spaces. Recently, a new class of sets namely ψ generalized semi-closed (briefly, ψgs- closed) sets were introduced in topological spaces and some of their basic properties were investigated. Nowadays, a new concept coined from topological spaces is the so-called bitopologi- cal spaces (briefly, BTS). Bose [2], studied semi continuity and semi open mappings in BTS. Thereafter, in [7] and [8], the concepts on generalized closed and semi open sets in bitopological spaces were investigated. The concepts of bitopological spaces have been widely investigated up to other types of spaces, like soft bitopological spaces. The researcher of this present study was inspired by the work of Şenel and Cagman where in [5] they studied soft closed sets on soft bitopological spaces. Thereafter in [6] they investigated soft topological subspaces. In addition, in [3] a new approach to Hausdorff space theory via the soft sets was investigated by Şenel and further studied soft topology generated by L-soft sets in [4]. With all these concepts in DOI: https://doi.org/10.29020/nybg.ejpam.v14i4.4060 Email address: lmernilotutanes@gmail.com (L.M.Tutanes) http://www.ejpam.com 1275 © 2021 EJPAM All rights reserved. L.M.Tutanes / Eur. J. Pure Appl. Math, 14 (4) (2021), 1275-1282 1276 mind, we are motivated to define and introduce ψgs-closed sets in bitopological spaces and will intend to further study in other spaces such as soft bitopological spaces. Moreover, we are interested to find the properties of ψgs-closed sets in BTS and their relationship to other existing sets and will intend to investigate the properties of ψgs- interior and ψgs-closure of a set. In general, this study establishes some properties of ψgs-closed set in bitopological spaces. Specifically, this study investigates some properties of ψgs-closed set in BTS; establishes the relationships between ψgs-closed set and other closed sets in BTS; and provides some properties of ψgs-closure and ψgs-interior in BTS. The major contributions of this study are the original results on ψgs-closed set in BTS. The findings reveal that every (i, j)-ψ-closed set, τi closed set, regular-closed set, semi-closed set, α-closed set, ψ-closed set, αgs-closed set in (X, τi) is (i, j)-ψgs-closed where i, j ∈ {1, 2}. Hence, (i, j)-ψgs-closed set is bigger than those of the mentioned sets. Also, it was found out that the intersection of (i, j)-ψgs-closed sets is (i, j)-ψgs-closed. Furthermore, the results on ψgs-closure and ψgs-interior in BTS are analogous to that in other spaces. We are motivated to have the results or theorems since these results could also be applied in other spaces to come up with analogous results or theorems. This study could serve as a resource material for future researches and possible applications. This may encourage other mathematics enthusiasts to come up with more results and to establish possible research directions for further study. 2. Preliminaries In this section, some basic definitions and some known results are provided. Examples are also given for a clearer understanding of several terms defined. A collection τ of subsets of a nonempty set X is a topology on X if ∅, X ∈ τ , {Mω : ω ∈ Ω} ⊆ τ implies ∪ω∈ΩMω ∈ τ , and A,B ∈ τ implies A ∩ B ∈ τ . If τ is a topology on X, then (X, τ) is called a topological space, and the elements of τ are called τ -open (or simply open) sets. A subset F of X is said to be τ -closed (or simply closed) if its complement X∖F is open. The interior of A, denoted by int(A), is the union of all open sets contained in A. That is, int(A) = ⋃ {O ∈ τ : O ⊆ A} . The closure of A, denoted by cl(A), is the intersection of all closed sets containing A. That is, cl(A) = ⋂ {F ⊆ X : F is closed and F ⊇ A} . Now, if τ1 and τ2 are arbitrary topologies on X then (X, τ1, τ2) is called a bitopological space. The interior of A and the closure of A with respect to τi are denoted by inti(A) and cli(A), respectively. Note that through out this context i, j ∈ {1, 2} such that i ̸= j. Definition 1. Let (X, τ) be a topological space. A subset A of X is called (i) semi-open set [12] if A ⊆ cl(int(A)); L.M.Tutanes / Eur. J. Pure Appl. Math, 14 (4) (2021), 1275-1282 1277 (ii) regular-open set [19] if A = int(cl(A)); (iii) α-open set [16] if A ⊆ int(cl(int(A))); (iv) semi-generalized closed (briefly, sg-closed) set [1] if scl(A) ⊆ U whenever A ⊆ U and U is semi-open in (X, τ); (v) αgs-closed set [18] if αcl(A) ⊆ U whenever A ⊆ U and U is semi-open in (X, τ); (vi) ψ-closed set [11] if scl(A) ⊆ U whenever A ⊆ U and U is sg-open in (X, τ); and (vii) ψ generalized semi-closed (briefly, ψgs-closed) set [9] if ψcl(A) ⊆ U whenever A ⊆ U and U is semi-open in (X, τ). The complement of semi-open (resp. regular-open, α-open, gs-closed, αgs-closed, ψ- closed, and ψgs-closed) set is called semi-closed (resp. regular-closed, α-closed, gs-open, αgs-open, ψ-open, and ψgs-open) set. Definition 2. Let (X, τ1, τ2) be a bitopological space. A subset A of X is called (i) (i, j)-semi open set [15] if A ⊆ clj(inti(A)); (ii) (i, j)-semi generalized closed (briefly, (i, j)-sg closed) set [17] if (i, j)-scl(A) ⊆ U whenever A ⊆ U and U is (i, j)-semi open; and (iii) (i, j)-ψ-closed set [20] if (i, j)-scl(A) ⊆ U whenever A ⊆ U and U is (i, j)-sg open. The complement of (i, j)-semi open (resp. (i, j)-sg closed and (i, j)-ψ-closed) set is called (i, j)-semi closed (resp. (i, j)-sg open and (i, j)-ψ-open) set. The next result was proven in [14]. Lemma 1. If a subset A of X is semi-open (respectively, semi-closed, sg-closed, gs-closed, g-closed, ψ-closed) set in (X, τi) for i ∈ {1, 2}, then it is semi-open (respectively, semi- closed, sg-closed, gs-closed, g-closed, ψ-closed) set in (X, τ1, τ2). The next Theorem is a composition of several results from Gowsalya, S. and Balamani, N. in [9]. Theorem 1. Let (X, τ1, τ2) be bitopological space. Then (i) Every semi-closed set in (X, τ) is ψgs-closed in (X, τ). (ii) Every closed set in (X, τ) is ψgs-closed set in (X, τ). (iii) Every regular-closed set in (X, τ) is ψgs-closed in (X, τ). (iv) Every α-closed set in (X, τ) is ψgs-closed in (X, τ). (v) Every ψ-closed set in (X, τ) is ψgs-closed in (X, τ). (vi) Every αgs-closed set in (X, τ) is ψgs-closed in (X, τ). L.M.Tutanes / Eur. J. Pure Appl. Math, 14 (4) (2021), 1275-1282 1278 3. ψgs-closed sets and its relationship to other closed sets in BTS In this section, some properties of ψgs-closed sets in BTS are investigated. Moreover, the relationship to some other existing closed sets in BTS is established. Definition 3. A subset A of a bitopological space (X, τ1, τ2) is called (i, j)-ψ generalized semi-closed (briefly, (i, j)-ψgs-closed) set if (i, j)-ψcl(A) ⊆ U whenever A ⊆ U and U is (i, j)-semi-open in (X, τ1, τ2), i, j ∈ {1, 2} where i ̸= j. The complement of (i, j)-ψgs-closed set is called (i, j)-ψgs-open set. Example 1. Let (X, τ1, τ2) be a bitopological space such that X = {a, b, c}, τ1 = {∅, X, {a}, {b}, {a, b}}, and τ2 = {∅, X, {a}, {a, b}, {a, c}} and A = {b, c}. Note that X is the only (1, 2)-semi open set containing A = {b, c}. Since A = {b, c} is a ψ closed set, it follows that (i, j)-ψcl({b, c}) = {b, c} ⊆ X. Thus A = {b, c} is a (1, 2)-ψ generalized semi closed set. Similarly, ∅, {b}, {c} and X are (1, 2)-ψ generalized semi closed sets. Throughout this context, the open (resp., closed) set in (X, τ1, τ2) is denoted by (i, j)- open (resp., (i, j)-closed) set. Proposition 1. Every (i, j)-ψ-closed set is (i, j)-ψgs-closed. Proof. Let A be (i, j)-ψ-closed set and U be (i, j)-semi-open in (X, τ1, τ2) such that A ⊆ U . Then (i, j)-ψcl(A) = A ⊆ U . Hence A is (i, j)-ψgs-closed in (X, τ1, τ2). Theorem 2. Every ψgs-closed set in (X, τi) is ψgs-closed set in (X, τ1, τ2). Proof. Let A be ψgs-closed set in (X, τi). Then ψcl(A) ⊆ U where U is semi-open in τi such that A ⊆ U . Since, for each i ∈ {1, 2}, every semi-open in (X, τi) is semi-open in (X, τ1, τ2) by Lemma 1, it follows that U is semi-open in (X, τ1, τ2). Moreover, by Lemma 1, every ψ-closed in τi is ψ-closed in (X, τ1, τ2). Thus (i, j)-ψcl(A) ⊆ ψcl(A) ⊆ U . Hence A is a ψgs-closed set in (X, τ1, τ2). The following corollary follows from Theorem 1 and Theorem 2. Corollary 1. Let (X, τi) be a topological space and (X, τ1, τ2) be bitopological space. Then the following statements hold. (i) Every τi closed set is (i, j)-ψgs-closed set. (ii) Every regular-closed set in (X, τi) is (i, j)-ψgs-closed. (iii) Every semi-closed set in (X, τi) is (i, j)-ψgs-closed. (iv) Every α-closed set in (X, τi) is (i, j)-ψgs-closed. (v) Every ψ-closed set in (X, τi) is (i, j)-ψgs-closed. (vi) Every αgs-closed set in (X, τi) is (i, j)-ψgs-closed. L.M.Tutanes / Eur. J. Pure Appl. Math, 14 (4) (2021), 1275-1282 1279 Theorem 3. Let A be a (i, j)-ψ-closed set and A ⊆ B ⊆ (i, j)-ψcl(A). Then B is also a (i, j)-ψgs-closed set. Proof. Let A be a (i, j)-ψ-closed set and A ⊆ (i, j)-ψcl(A). Suppose U is (i, j)-semi- open such that B ⊆ U . We want to show (i, j)-ψcl(B) ⊆ U . Since A ⊆ B and B ⊆ U , it follows that A ⊆ U . Also, by Proposition 1, A is a (i, j)-ψgs-closed set since A is a (i, j)-ψ-closed set; and so (i, j)-ψcl(A) ⊆ U . Note that B ⊆ (i, j)-ψcl(A) implies (i, j)-ψcl(B) ⊆ (i, j)-ψcl((i, j)-ψcl(A)) = (i, j)-ψcl(A) ⊆ U. Theorem 4. Let {Ak|Ak is (i, j)-ψgs-closed set, k ∈ N}. Then ⋂∞ k=1Ak is (i, j)-ψgs- closed set. Proof. Suppose U = ⋂∞ k=1 Uk is (i, j)-semi-open such that ⋂∞ k=1Ak ⊆ U . We want to show that (i, j)-ψcl( ⋂∞ k=1Ak) ⊆ U . From our assumption, Ak is (i, j)-ψgs-closed set for each k ∈ N. It follows that (i, j)-ψcl(Ak) ⊆ Uk for each k such that Ak ⊆ Uk where Uk is (i, j)-semi-open. Now, (i, j)-ψcl( ∞⋂ k=1 Ak) ⊆ ∞⋂ k=1 (i, j)-ψcl(Ak) ⊆ ∞⋂ k=1 Uk = U. Note that if Ak is (i, j)-ψgs-closed set, then by Definition 3, X∖Ak is (i, j)-ψgs-open set. Moreover, by De Morganś law, X∖( ⋂∞ k=1Ak) = ⋃∞ k=1(X∖Ak). Hence the following corollary follows. Corollary 2. If {Bk|Bk is (i, j)-ψgs-open set, k ∈ N}, then ⋃∞ k=1Bk is (i, j)-ψgs-open set. 4. ψgs-closure and ψgs-interior in BTS In this section, the ψgs-closure and ψgs-interior in bitopological spaces are introduced and some of their properties are explored. Definition 4. Let (X, τ1, τ2) be a bitopological space and A ⊆ X. An element x ∈ A is called (i, j)-ψgs-interior point of A if there exists a (i, j)-ψgs-open set O such that x ∈ O ⊆ A. The set of all (i, j)-ψgs-interior points of A is called the (i, j)-ψgs-interior of A and is denoted by (i, j)-ψgs-Int(A). Theorem 5. The ψgs-interior of a subset A of X is the countable union of ψgs-open sets contained in A, that is, (i, j)-ψgs-Int(A) = ∪{O : O is (i, j)-ψgs-open and O ⊆ A}. L.M.Tutanes / Eur. J. Pure Appl. Math, 14 (4) (2021), 1275-1282 1280 Proof. Let x ∈ (i, j)-ψgs-Int(A). Then there exists a (i, j)-ψgs-open set O such that x ∈ O ⊆ A implying x ∈ ∪{O : O is (i, j)-ψgs-open and O ⊆ A}. Next, suppose y ∈ ∪{O : O is (i, j)-ψgs-open and O ⊆ A}. Then there exists (i, j)-ψgs-open set O0 ⊆ A such that y ∈ O0. Thus y ∈ (i, j)-ψgs-Int(A). The previous theorem implies that (i, j)-ψgs-Int(A) is contained in A, where A is any subset of X, being the union of all (i, j)-ψgs-open sets contained in A. Now Corollary 2 entails that the arbitrary union of (i, j)-ψgs-open sets is also (i, j)-ψgs-open, hence we can say that (i, j)-ψgs-Int(A) is (i, j)-ψgs-open. Consequently, (i, j)-ψgs-Int(A) is the largest (i, j)-ψgs-open set contained in A, as stated in the following remark. Remark 1. Let (X, τ1, τ2) be a bitopological space and A,B ⊆ X. Then the following hold: (i) (i, j)-ψgs-Int(A) ⊆ A; (ii) (i, j)-ψgs-Int(A) is (i, j)-ψgs-open set; and (iii) If B ⊆ A such that B is (i, j)-ψgs-open set, then B ⊆ (i, j)-ψgs-Int(A). Theorem 6. Let (X, τ1, τ2) be a bitopological space and A ⊆ X. A is (i, j)-ψgs-open set, if and only if (i, j)-ψgs-Int(A) = A. Proof. Let A be (i, j)-ψgs-open set and x ∈ A. Note that by Remark 1 (i), (i, j)-ψgs-Int(A) ⊆ A. Hence it suffices to show A ⊆ (i, j)-ψgs-Int(A). Suppose x /∈ (i, j)-ψgs-Int(A) . Then x /∈ O for all (i, j)-ψgs-open sets O such that A ⊆ O. It follows that x ∈ X∖O such that A∩(X∖O) = ∅. This is a contradiction since x ∈ A∩X∖O. Thus x ∈ (i, j)-ψgs-Int(A) . Consequently, (i, j)-ψgs-Int(A) = A. Conversely, suppose (i, j)-ψgs-Int(A) = A. By Remark 1 (ii), (i, j)-ψgs-Int(A) is (i, j)-ψgs-open set, and so A is (i, j)-ψgs-open set. Definition 5. Let A ⊆ X. Then x ∈ X is (i, j)-ψgs-adherent to A if V ∩A ̸= ∅ for every (i, j)-ψgs-open set V containing x. The set of all (i, j)-ψgs-adherent points of A is called the (i, j)-ψgs-closure of A and is denoted by (i, j)-ψgs-Cl(A). Theorem 7. The (i, j)-ψgs-closure of a subset A of X is the countable intersection of (i, j)-ψgs-closed sets containing A, that is, (i, j)-ψgs-Cl(A) = ⋂ {F : F is (i, j)-ψgs-closed and A ⊆ F}. Proof. Let x ∈ (i, j)-ψgs-Cl(A). Then O ∩ A ̸= ∅ for every (i, j)-ψgs-open set O containing x. Suppose x /∈ ∩{F : F is (i, j)-ψgs-closed and A ⊆ F}. It follows that there exists (i, j)-ψgs-closed F0 such that A ⊆ F0 and x ∈ X∖F0. Note that X∖F0 is (i, j)-ψgs-open set containing x such that (X∖F0) ∩A = ∅, a contradiction. Thus x ∈ ∩{F : F is (i, j)-ψgs-closed and A ⊆ F}. REFERENCES 1281 Next, let y ∈ ∩{F : F is (i, j)-ψgs-closed and A ⊆ F}. Then y ∈ F for all (i, j)-ψgs- closed such that A ⊆ F . Suppose on the contrary, y /∈ (i, j)-ψgs-Cl(A). It implies that U ∩ A = ∅ for some (i, j)-ψgs-closed set U containing y. Hence there exists (i, j)-ψgs- closed set X∖U such that y /∈ X∖U and A ⊆ X∖U , a contradiction. Consequently, y ∈ (i, j)-ψgs-Cl(A). Theorem 7 indicates that (i, j)-ψgs-Cl(A) contains A since the intersection of all (i, j)-ψgs-closed sets contains A. Now Theorem 4 implies that the arbitrary intersection of (i, j)-ψgs-closed sets is also (i, j)-ψgs-closed, thus it follows that (i, j)-ψgs-Cl(A) is (i, j)-ψgs-closed. Hence, (i, j)-ψgs-Cl(A) is the smallest (i, j)-ψgs-closed set that contains A, as stated in the following remark. Remark 2. Let (X, τ1, τ2) be a bitopological space and A,B ⊆ X. Then the following hold: (i) A ⊆ (i, j)-ψgs-Cl(A); (ii) (i, j)-ψgs-Cl(A) is (i, j)-ψgs-closed set; and (iii) If A ⊆ B such that B is (i, j)-ψgs-closed set, then (i, j)-ψgs-Cl(A) ⊆ B. Theorem 8. A is (i, j)-ψgs-closed set, if and only if (i, j)-ψgs-Cl(A) = A. Proof. Let A be (i, j)-ψgs-closed set and x ∈ (i, j)-ψgs-Cl(A). Then for all (i, j)-ψgs- open set U containing x, we have U ∩ A ̸= ∅. Suppose on the contrary, x /∈ A. Then x ∈ X∖A where X∖A is (i, j)-ψgs-open and (X∖A) ∩ A = ∅, a contradiction since x ∈ (i, j)-ψgs-Cl(A). Thus x ∈ A, and so (i, j)-ψgs-Cl(A) ⊆ A. Note that by Re- mark 2 (i), A ⊆ (i, j)-ψgs-Cl(A), and hence (i, j)-ψgs-Cl(A) = A. Conversely, suppose (i, j)-ψgs-Cl(A) = A. By Remark 2 (ii), (i, j)-ψgs-Cl(A) is (i, j)-ψgs-closed set, and so A is (i, j)-ψgs-closed set. Acknowledgements We are thankful to the Bukidnon State University Research Unit for the financial assistance. References [1] P. Bhattacharyya and B.K. Lahiri. Semi-generalized closed sets in topology. Indian J. Math., 29:376–382, 1987. [2] S. Bose. Semi Open Sets, Semi Continuity and Semi Open Mappings in Bitopological Spaces. Bull. Cal. Math. Soc., 73:237–246, 1981. [3] G. Şenel. A New Approach to Hausdorff Space Theory via the Soft Sets. Mathematical Problems in Engineering, 9:1–6, 2016. REFERENCES 1282 [4] G. Şenel. Soft Topology Generated by L-Soft Sets. Journal of New Theory, 4(24):88– 100, 2018. [5] G. Şenel and N. Cagman. Soft Closed Sets on Soft Bitopological Space. Journal of New Results in Science, 3(5):57–66, 2014. [6] G. Şenel and N. Cagman. Soft topological subspaces. Annals of Fuzzy Mathematics and Informatics, 10(4):525–535, 2015. [7] T. Fukutake. On Generalized Closed Sets in Bitopological spaces. Bull. Fukuoka. Univ. of Educ., 35:19–28, 1985. [8] T. Fukutake. Semi Open Sets on Bitopological Spaces. Bull. Fukuoka. Univ. of Educ., 38:1–7, 1989. [9] S. Gowsalya and N. Balamani. ψgs-Closed Sets in Topological Spaces. International Journal of Advance Foundation and Research in Computer, 3(4):52–61, 2016. [10] M. Veera kumar. Between closed and g-closed sets. Mem.Fac Sci. KochiUniv.Math., (21):1–19, 2000. [11] M. Veera kumar. Between ψ-closed sets and gsp-closed sets spaces. Antarctica. J.Math., 2(1):123–141, 2005. [12] N. Levine. Semi-open sets and semi-continuity in topological spaces. Amer. Math. Monthly, (70):36–41, 1963. [13] N. Levine. Generalized closed sets in topological spaces. Rend. Circ. Mat. Palermo, 19(2):89–96, 1970. [14] Y. Mahdi. Semi-open and semi-closed sets in bitopological spaces. In First Science Conference of Education College, pages 18–19, Hillah, 2007. Babylon Univ. [15] Y. K. Mahdi. Semi-open and semi-closed set in Bitopological Spaces. The first scientific conference of the Faculty of Physical Education, 18:1–8, 2007. [16] O. Njastad. On some classes of nearly open sets. Pacific J. Math., 15:961–970, 1965. [17] H.M. Abu-Donia O.A. El-Tantawi. Generalized separation axioms in bitopological spaces. Arab. J. Sci. Eng., 1:117–129, 2005. [18] M. Rajamani and K. Vishwanathan. αgs-closed sets in topological spaces. Acta Cienia Indica, XXXM(3):521–526, 2004. [19] M. Stone. Application of the theory of Boolean rings to general topology. Trans. Amer.Math. Soc., 41:374–481, 1937. [20] R. Nithya kalyani Veronica Viayan. A study on (i, j)-ψ⋆, and (i, j)-ψ closed sets in bitopological spaces. International Journal of Computer Application, 4:40–48, 2013.