EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 4, 2023, 2049-2065 ISSN 1307-5543 – ejpam.com Published by New York Business Global Generalized Dense Sets in Bigeneralized Topological Spaces Yasser Farhat1, Vadakasi Subramanian2,∗ 1 Academic Support Department, Abu Dhabi Polytechnic, P. O. Box 111499, Abu Dhabi, United Arab Emirates 2 Department of Mathematics, A.K.D.Dharma Raja Women’s College, Rajapalayam, India Abstract. In this article, in a bigeneralized topological space, we introduce an interesting tool namely, (s, v)-dense set, and examine its significance of this set. Also, we give the relationships among nowhere-dense sets defined in both generalized and bigeneralized topological space and give some of their properties by using functions. Finally, we give some applications for (s, v)-dense and (s, v)-nowhere dense sets in a soft set theory. 2020 Mathematics Subject Classifications: 54A05, 54A10 Key Words and Phrases: Bigeneralized topological spaces, µ(s,v)-open, µ(s,v)-closed, µ(s,v)- dense, g(s,v)-continuous function. 1. Introduction In [2], Császár defined the notion of generalized topological space. Some researchers have found various new concepts in this space and examined their nature in a generalized topological space. Especially, nowhere dense and dense sets were introduced by Ekici in a generalized topological space [6]. He has given few results for nowhere-dense and dense sets in a generalized topological space. Some researchers proved various properties for nowhere dense sets e.g. [9, 12, 14]. In- spired by this, Korczak-Kubiak, et al. introduced two new generalized topologies, namely, µ⋆ and µ⋆⋆; then examined the nature of nowhere dense set using µ⋆ and µ⋆⋆ [8]. In [7], J.C. Kelly introduced the notion of bitopological space. Motivated by this, C. Boonpok introduced the concept of bigeneralized topological space in 2010 [1]. He proved some results about (m,n)-closed sets in bigeneralized topological space. In this paper, we define the generalization of dense sets, namely, (s, v)-dense in a bigeneralized topological space. In a bigeneralized topological space, various properties for (s, v)-dense and (s, v)-nowhere dense sets are launched. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i4.4911 Email addresses: farhat.yasser.1@gmail.com (Y. Farhat), vadakasivigneswaran@gmail.com (S.Vadakasi) https://www.ejpam.com 2049 © 2023 EJPAM All rights reserved. Y. Farhat, V. Subramanian / Eur. J. Pure Appl. Math, 16 (4) (2023), 2049-2065 2050 The basic definitions and results are presented in section 2 which is useful for the development of the following sections. In section 3, in a bigeneralized topological space, new results for (s, v)-dense sets are proven. The necessary conditions for a given set is (s, v)-dense are given. Section 4, some properties for (s, v)-nowhere dense sets are proven. In a bigeneralized topological space, the relationship between µ-nowhere dense and (s, v)- nowhere dense sets are examined. Finally, the set (s, v)-codense is defined and find few results for this set. In section 5, the nature of (s, v)-dense and (s, v)-codense sets are examined by func- tions in a bigeneralized topological space. In the last section, we define a soft set using (s, v)-dense, (s, v)-nowhere dense, and (s, v)-codense sets are defined in a bigeneralized topological space. 2. Preliminaries Let µ be the collection of subsets of a non-null set X. µ is called generalized topol- ogy [2] in X if it contains the empty set and is closed under arbitrary union. Then (X,µ) is called generalized topological space (GTS) [2]. If µ contains X, then (X,µ) is called as a strong generalized topological space (sGTS) [9]. In, [3], let Q be the subset of (X,µ), • If Q ∈ µ, then Q is called µ-open. • If X −Q ∈ µ, then Q is said to be µ-closed. • The interior of Q denoted by iµQ, is the union of all µ-open sets contained in Q. • The closure of Q denoted by cµQ, is the intersection of all µ-closed sets containing Q. For ease of notation, we write i(Q) and c(Q) when no confusion can arise. Korczak - Kubiak, et.al [8] defined the following notations; µ̃ = {L ∈ µ | L ̸= ∅}. µ(x) = {L ∈ µ | x ∈ L}. Let Q be a subset of a generalized topological space (X,µ). Then Q is said to be ; • µ-nowhere dense [6] if ic(Q) = ∅ ; • µ-dense [6] if cQ = X ; • µ-codense [5] if c(X −Q) = X. Let µ1, µ2 be two GT in a non-null set X. Then (X,µ1, µ2) is called as a bigeneralized topological space (BGTS) [1]. Let (X,µ1, µ2) be a BGTS, D ⊂ X. T The closure of D is notated by cs(D) and is(D) denote the interior of D with respect to µs, respectively, for s = 1, 2 [1]. In a BGTS (X,µ1, µ2), let Q,P ⊂ X. Then • Q is called (s, v)-closed [1] if cs(cv(Q)) = Q, where s, v = 1 or 2 ; s ̸= v. Y. Farhat, V. Subramanian / Eur. J. Pure Appl. Math, 16 (4) (2023), 2049-2065 2051 • If X −Q is (s, v)-closed, then Q is called (s, v)-open [1] where s, v = 1 or 2 ; s ̸= v. • P is called µ(s,v)-closed [4] if cµv(P ) ⊂ K whenever P ⊂ K and K is µs-open in X, for s, v = 1, 2 ; s ̸= v. • If X − P is µ(s,v)-closed, then P is called µ(s,v)-open [4] where s, v = 1 or 2 ; s ̸= v. In [1], a subset Q of a BGTS (X,µ1, µ2) is called • (s, v)-µ-regular open if Q = is(cv(Q)) for s, v = 1 or 2 ; s ̸= v. • (s, v)-µ-semi-open if Q ⊆ cv(is(Q)) for s, v = 1 or 2 ; s ̸= v. • (s, v)-µ-preopen if Q ⊆ is(cv(Q)) for s, v = 1 or 2 ; s ̸= v. • (s, v)-µ-α-open if Q ⊆ is(cv(is(Q))) for s, v = 1 or 2 ; s ̸= v. Lemma 1. [Proposition 3.4, [1]] Let K be a subset of a BGTS (X,µ1, µ2). Then K is (s, v)-closed ⇔ K is both µ-closed in (X,µs) and (X,µv) where s, v = 1 or 2 ; s ̸= v. Lemma 2. [Proposition 3.3, [4]] Let (X,µ1, µ2) be a BGTS, K ⊂ X. Then K is µ(s,v)- closed where s, v = 1, 2 ; s ̸= v whenever K is µv-closed. Lemma 3. [Lemma 3.2, [9]] Let D,K be two subsets of a generalized topological space (X,µ). If K ∈ µ̃ and K ∩D = ∅, then K ∩ cD = ∅. Lemma 4. [Proposition 3.3, [9]] In a GTS (X,µ), Q ∈ D(µ) ⇔ H ∩Q ̸= ∅ for any H ∈ µ̃ where D(µ) = {P ⊂ X | cµ(P ) = X}. Lemma 5. [Proposition 2.2, [10]] Let P,Q be two subsets of a GTS (X,µ). Then the followings are true: (a) cµ(X − P ) = X − iµ(P ) ; iµ(X − P ) = X − cµ(P ). (b) If (X − P ) ∈ µ, then cµ(P ) = P and if P ∈ µ, then iµ(P ) = P. (c) If P ⊆ Q, then cµ(P ) ⊆ cµ(Q) and iµ(P ) ⊆ iµ(Q). (d) P ⊆ cµ(P ) and iµ(P ) ⊆ P. (e) cµ(cµ(P )) = cµ(P ) and iµ(iµ(P )) = iµ(P ). 3. Nature of (s, v)-dense sets Here, we define a generalized dense set using two generalized topologies namely, (s, v)-dense set, and analyze its nature in a BGTS (X,µ1, µ2). Definition 1. Let D be a non-null subset of a bigeneralized topological space (X,µ1, µ2). Then D is called (s, v)-dense if cs(cv(D)) = X where s, v = 1, 2 and s ̸= v. Moreover, (s, v)−D(X) = {Q ⊂ X | Q is (s, v)-dense in X} for s, v = 1, 2 ; s ̸= v. Example 2. Consider the BGTS (X,µ1, µ2) where X = {e, f, k, l}; µ1 = {∅, {e}, {e, f}, {f, k}, {e, f, k}} and Y. Farhat, V. Subramanian / Eur. J. Pure Appl. Math, 16 (4) (2023), 2049-2065 2052 µ2 = {∅, {e, f}, {f, l}, {e, f, l}}. Then (s, v)−D(X) = {Q ⊂ X | either e ∈ Q or f ∈ Q} where s, v = 1, 2 ; s ̸= v. In a GTS, every superset of a (s, v)-dense set is (s, v)-dense where s, v = 1, 2 and s ̸= v. Theorem 3. Let (X,µ1, µ2) be a BGTS and Q be a non-null subset of X. Then Q is (s, v)-dense ⇔ cvQ ∩H ̸= ∅ for every H is a non-null µs-open set where s, v = 1, 2 and s ̸= v. Proof. Suppose Q ∈ (s, v) − D(X) for s, v = 1, 2 ; s ̸= v, then cs(cv(Q)) = X and so X − (cs(cv(Q))) = ∅ where s, v = 1, 2 and s ̸= v. By Lemma 5, X − (cs(cv(Q))) = is(X − (cv(Q))), so that is(X − (cv(Q))) = ∅ which implies that cv(Q) ∩ H ̸= ∅ for every H is a non-null µs-open set where s, v = 1, 2 and s ̸= v. Conversely, assume that, cv(Q) ∩ H ̸= ∅ for every H is a non-null µs-open set where s, v = 1, 2 and s ̸= v. Then is(X−(cv(Q))) = ∅ and so cs(cv(Q)) = X, by Lemma 5 where s, v = 1, 2 and s ̸= v. Hence Q is (s, v)-dense for s, v = 1, 2 and s ̸= v. Theorem 4 and Example 5 are described in the below diagram. µs − dense (s, v)− dense µv − dense / / Theorem 4. In a BGTS (X,µ1, µ2), if K is either µs-dense or µv-dense, then K is (s, v)-dense where s, v = 1, 2 ; s ̸= v. Proof. Assume that, K is µs-dense where for s = 1, 2. Then cs(K) = X for s = 1, 2. Take s = 2 and v = 1. Then K is µ2-dense. Since K ⊂ c1(K) we have c2(K) ⊂ c2(c1(K)). Hence K ∈ (2, 1)−D(X) (1) Take s = 1 and v = 2. Then K is µ1-dense. Since K ⊂ c2(K) we have c1(K) ⊂ c1(c2(K)). Thus, K ∈ (1, 2)−D(X) (2) From (1) & (2), K is (s, v)-dense where s, v = 1, 2 and s ̸= v. Similarly, we can prove that K is (s, v)-dense if K is µv-dense where s, v = 1, 2 and s ̸= v. Example 5 describes that the Theorem 4 is not reversible. Generally, (1, 2)−D(X) ̸= (2, 1)−D(X) in a bigeneralized topological space as given in Example 6. Example 5. Consider the bigeneralized topological space (X,µ1, µ2), X = {e, f, k, l}; µ1 = {∅, {e, l}, {f, l}, {e, f, l}} and Y. Farhat, V. Subramanian / Eur. J. Pure Appl. Math, 16 (4) (2023), 2049-2065 2053 µ2 = {∅, {e, k}, {f, k}, {e, f, k}}. Here {k} is (2, 1)-dense. But {k} is not µ1-dense. Also, {l} is (1, 2)-dense. But {l} is not µ2-dense. Example 6. Consider the bigeneralized topological space (X,µ1, µ2) whereX = {e, f, k, l}; µ1 = {∅, {e, f}, {f, k}, {e, f, k}} and µ2 = {∅, {e}, {e, l}, {k, l}, {e, k, l}}. Then • (1, 2)−D(X) = {{e}, {f}, {k}, {l}, {e, f}, {e, k}, {e, l}, {f, k}, {f, l}, {k, l}, {e, f, k}, {e, f, l}, {e, k, l}, {f, k, l}, X}. • (2, 1)−D(X) = {{e}, {f}, {e, f}, {e, k}, {e, l}, {f, k}, {f, l}, {e, f, k}, {e, f, l}, {e, k, l}, {f, k, l}, X}. Thus, (1, 2)−D(X) ̸= (2, 1)−D(X). Theorem 7. Let µ1 and µ2 be two generalized topologies in X. If µs ⊆ µv, then (v, s) − D(X) ⊆ (s, v)−D(X) where s, v = 1, 2 and s ̸= v. Proof. We give the detailed proof only for s = 1 and v = 2. Suppose that µ1 ⊆ µ2 and Q ∈ (2, 1) − D(X), then c2(c1(Q)) = X. By Lemma 4, c1(Q) ∩ H ̸= ∅ for every H ∈ µ̃2. Take G ∈ µ̃1 we get G ∈ µ̃2 for that c1(Q) ∩ G ̸= ∅. Since Q ⊂ c2(Q) we have c1(Q) ⊂ c1(c2(Q)). Thus, c1(c2(Q)) ∩ G ̸= ∅. Since G is an arbitrary non-null µ1-open set we have c1(c1(c2(Q))) = X, by Lemma 4. Hence c1(c2(Q)) = X, by Lemma 5(e). Therefore, Q ∈ (1, 2)−D(X). Theorem 8. Let (X,µ1, µ2) be a BGTS and D be a non-null subset of X. If D ∈ (s, v)− D(X), then D ∩ H ̸= ∅ for every H is a non-null (s, v)-open set in X for s, v = 1, 2 ; s ̸= v. Proof. Take s = 1 and v = 2. Assume that, D is (1, 2)-dense. Then c1(c2(D)) = X. Let H be a non-null (1, 2)-open set. By Lemma 1, H ∈ µ̃1 (3) H ∈ µ̃2 (4) Then c2(D)∩H ̸= ∅, by Lemma 4 and (3). From (4) and c2(D)∩H ̸= ∅ we have D∩H ̸= ∅, by Lemma 3. Thus, D ∩H ̸= ∅ for every H is a non-null (1, 2)-open set. Take s = 2 and v = 1. By similar considerations in the above case, we get the proof. Theorem 9. Let (X,µ1, µ2) be a BGTS, D ⊂ X. If D ∩ H ̸= ∅ for every H ̸= ∅ is µ(s,v)-open, then D ∈ (s, v)−D(X); s, v = 1, 2 and s ̸= v. Proof. We give the detailed proof for s = 1 and v = 2 only. Suppose that D ∩H ̸= ∅ for every H is non-null µ(1,2)-open. By Theorem 4, we have to prove D is µ2-dense. Let B ∈ µ̃2. Then B is a non-null µ(1,2)-open set in X, by Lemma 2. By assumption, D ∩B ̸= ∅. Therefore, D is a µ2-dense set. Hence D is a (1, 2)-dense set. Y. Farhat, V. Subramanian / Eur. J. Pure Appl. Math, 16 (4) (2023), 2049-2065 2054 The below Example 10 describes that the converse part of Theorem 9 is generally not true. Example 10. Take X = {e, f, k, l}; µ1 = {∅, {e, f}, {f, l}, {e, f, l}} and µ2 = {∅, {e, k}, {f, k}, {e, f, k}}. Then µ(1,2) = {∅, {e}, {f}, {l}, {e, f}, {e, k}, {e, l}, {f, k}, {f, l}, {e, f, k}, {e, f, l}} and µ(2,1) = {∅, {e}, {f}, {k}, {e, f}, {e, k}, {f, k}, {f, l}, {e, f, k}, {e, f, l}, {f, k, l}}. Take P = {e}. Then P ∈ (1, 2) − D(X). But P ∩ Q = ∅ where Q = {l} is a non-null µ(1,2)-open set. Let M = {f} ⊂ X. Then M ∈ (2, 1) − D(X). But M ∩ L = ∅ where L = {e} is a non-null µ(2,1)-open set. Q ∈ µ̃s Q is (s, v)− µ− semi open. Q is (s, v)− µ− preopen Q is (s, v)− µ− α− open The following Lemma 6 describes the above diagram. Lemma 6. Let (X,µ1, µ2) be a BGTS. If Q ∈ µ̃s, then the below results are true. (a) Q is (s, v)-µ-semi open. (b) Q is (s, v)-µ-preopen. (c) Q is (s, v)-µ-α-open where s, v = 1, 2 and s ̸= v. Proof. We give the detailed proof for (b) only. Suppose that, Q ∈ µ̃s for s = 1, 2. Then is(Q) = Q for s = 1, 2. Since Q ⊂ cv(Q) for v = 1, 2 we have is(Q) ⊂ is(cv(Q)) where s, v = 1, 2 and s ̸= v. Thus, Q ⊂ is(cv(Q)) where s, v = 1, 2 and s ̸= v. Hence Q is a (s, v)-µ-preopen set in X for s, v = 1, 2 ; s ̸= v. Theorem 11. Let (X,µ1, µ2) be a BGTS. Then D ∈ (s, v) − D(X) if any one of the following is true. (a) D ∩M ̸= ∅ for every M is a non-null (s, v)-µ-semi open set in X (b) D ∩M ̸= ∅ for every M is a non-null (s, v)-µ-preopen set in X (c) D ∩M ̸= ∅ for every M is a non-null (s, v)-µ-α-open set in X where s, v = 1, 2; s ̸= v. Proof. We give the detailed proof for (b) only. Suppose that D ∩M ̸= ∅ for every M is a non-null (s, v)-µ-preopen set in X where s, v = 1, 2 and s ̸= v. It is enough to prove, D is µs-dense set in X for s = 1, 2, by Theorem 4. Let B ∈ µ̃s for s = 1, 2. By Lemma 6, B is a non-null (s, v)-µ-preopen set in X where s, v = 1, 2 and s ̸= v. By assumption, D ∩ B ̸= ∅. Therefore, D is a µs-dense set for s = 1, 2. Hence D is (s, v)-dense where s, v = 1, 2 and s ̸= v. Y. Farhat, V. Subramanian / Eur. J. Pure Appl. Math, 16 (4) (2023), 2049-2065 2055 Example 12 explains that the reverse part of Theorem 11 is generally not true. Example 12. (a) Consider the bigeneralized topological space (X,µ1, µ2) where X = {e, f, k, l, r}; µ1 = {∅, {e, f}, {e, l}, {f, l}, {e, f, l}} and µ2 = {∅, {e, f, k}, {e, f, l}, {e, k, r}, {e, f, k, l}, {e, f, k, r}, X}. Take A = {k, l, r}. Then A is (1, 2)-dense set. But A ∩ G = ∅ where G = {e, f} is a non-null µ(1,2)-µ-semi open set. Let B = {l, r} ⊂ X. Then B is (2, 1)-dense set. But B ∩H = ∅ where H = {e, f, k} is a non-null µ(2,1)-µ-semi-open set. (b) Consider the BGTS (X,µ1, µ2), X = [0, 3]; µ1 = {∅, [0, 2), (1, 3], [0, 3]} and µ2 = {∅, [0, 32 ], (1, 2], [0, 2]}. Let A = (0, 1)∪ (32 , 3]. Then A ∈ (s, v)−D(X) where s, v = 1, 2 and s ̸= v. But A∩B = ∅ where B = {3 2} is a non-null (s, v)-µ-preopen set in X where s, v = 1, 2 ; s ̸= v. (c) Consider the BGTS (X,µ1, µ2), X = [0, 4]; µ1 = {∅, [0, 2), (1, 2)} and µ2 = {∅, [0, 2), (1, 2], (1, 3), [0, 2], [0, 3)}. Let P = (0, 1) ∪ [2, 4]. Then P ∈ (1, 2) − D(X). But P ∩ Q = ∅ where Q = [1, 2) is a non-null (s, v)-µ-α-pen set in X where s, v = 1, 2 and s ̸= v. Let C = (0, 1) ∪ [3, 4]. Then C is (2, 1)-dense set in X. But C ∩D = ∅ where D = [1, 3) is a non-null (s, v)-µ-α-pen set in X where s, v = 1, 2 and s ̸= v. 4. Generalized nowhere dense sets Here, we find the new results for (s, v)-nowhere dense set in a BGTS. Definition 13. [13] Let (X,µ1, µ2) be a BGTS and D ⊂ X. Then D is called (s, v)- nowhere dense if is(cv(D)) = ∅ where s, v = 1, 2 and s ̸= v. We notated, (s, v)−N (X) = {Q ⊂ X | Q is (s, v)-nowhere dense inX} where s, v = 1, 2 ; s ̸= v. Y. Farhat, V. Subramanian / Eur. J. Pure Appl. Math, 16 (4) (2023), 2049-2065 2056 Example 14. Take X = {e, f, k, l}; µ1 = {∅, {e, f}, {e, k}, {e, f, k}} and µ2 = {∅, {e, l}, {f, l}, {e, f, l}}. Then {k} is a non-null (s, v)-nowhere dense set in (X,µ1, µ2) where s, v = 1, 2 ; s ̸= v. In a bigeneralized topological space, if Q ∈ (s, v) − N (X) and P ⊂ Q, then P ∈ (s, v)−N (X) where s, v = 1, 2 and s ̸= v. Theorem 15. In a BGTS (X,µ1, µ2), D ∈ (s, v)−N (X) if and only if cv(D) ∈ (s, v)− N (X) where s, v = 1, 2 and s ̸= v. In a BGTS (X,µ1, µ2), (1, 2)−N (X) ̸= (2, 1)−N (X) as shown by the below Example 16 . Also, this example shows that (s, v)−N (X) is not closed under finite union in general. Example 16. Let (X,µ1, µ2) be a BGTS where X = {e, f, k, l}; µ1 = {∅, {e, l}, {f, l}, {e, f, l}} and µ2 = {∅, {e, f}, {f, l}, {e, f, l}}. Then • (1, 2)−N (X) = {∅, {e}, {k}, {l}, {e, k}, {k, l}} • (2, 1)−N (X) = {∅, {e}, {f}, {k}, {e, k}, {f, k}}. Thus, (2, 1)−N (X) ̸= (1, 2)−N (X). Here {e} and {l} are in (1, 2)−N (X). But {e, l} /∈ (1, 2)−N (X). Also, {e} and {f} are in (2, 1)−N (X). But {e, f} /∈ (2, 1)−N (X). Theorem 17. Let µ1 and µ2 be two generlized topologies on a non-null set X. If µs ⊆ µv, then (v, s)−N (X) ⊆ (s, v)−N (X) where s, v = 1, 2 and s ̸= v. Proof. We give the detailed proof only for s = 1 and v = 2. Assume that, µ1 ⊆ µ2 (5) Let D ∈ (2, 1)−N (X). Then i2(c1(D)) = ∅. Suppose i1(c2(D)) ̸= ∅. There exists K ∈ µ̃1 such that K ⊂ c2(D). From (5), K ∈ µ̃2. Then i2(c2(D)) ̸= ∅. By (5) we get c2(D) ⊂ c1(D). Thus, i2(c1(D)) ̸= ∅ which is not possible. Therefore, i1(c2(D)) = ∅. Hence D ∈ (1, 2)−N (X). The following Theorem 19 describes the below diagram. Y. Farhat, V. Subramanian / Eur. J. Pure Appl. Math, 16 (4) (2023), 2049-2065 2057 µv − nowhere dense (s, v)− nowhere dense µs − nowhere dense The following Example 18 shows that the existence of the below Theorem 19. Example 18. (a) Fix s = 1, v = 2. Consider the bigeneralized topological space (X,µ1, µ2) where X = {p, q, r, s}; µ1 = {∅, {p, r}, {q, r}, {p, q, r}} and µ2 = {∅, {p, r}, {q, r}, {p, s}{p, q, r}, {p, r, s}, X}. Obviously, µ1 ⊂ µ2. Take K = {p, s} and L = {q}. Then K is a µ1-nowhere dense set and L is a µ2-nowhere dense set. Here, both K and L are in (1, 2)−N (X). (b) Fix s = 2, v = 1. Consider the bigeneralized topological space (X,µ1, µ2) where X = {p, q, r, s}; µ1 = {∅, {p, s}, {r, s}, {q, s}{p, q, s}, {p, r, s}, {q, r, s}, X} and µ2 = {∅, {q, s}, {r, s}, {q, r, s}}. Clearly, µ2 ⊂ µ1. Take H = {r} and D = {p, r}. Then H is a µ1-nowhere dense set and D is a µ2-nowhere dense set. Also, both H and D are in (2, 1)−N (X). Theorem 19. Let µ1, µ2 be two generlized topologies on X and µs ⊆ µv where s, v = 1, 2 and s ̸= v. If P ⊂ X is µv-nowhere dense set or µs-nowhere dense set, then P ∈ (s, v)− N (X) where s, v = 1, 2 and s ̸= v. Proof. We give the detailed proof only for s = 2 and v = 1. Assume that, µ2 ⊆ µ1 (6) Let P be a µ1-nowhere dense set. Then i1(c1(P )) = ∅. Suppose i2(c1(P )) ̸= ∅. Then there is Q ∈ µ̃2 such that Q ⊂ c1(P ). From (6), Q ∈ µ̃1. Then i1(c1(P )) ̸= ∅ which is not possible. Therefore, i2(c1(P )) = ∅. Hence P ∈ (2, 1)−N (X). Let P be a µ2-nowhere dense set. Then i2(c2(P )) = ∅. Suppose i2(c1(P )) ̸= ∅. Then there is a set M ∈ µ̃2 such that M ⊂ c1(P ). By (6), i2(c2(P )) ̸= ∅ which is not possible. Therefore, i2(c1(P )) = ∅. Hence P ∈ (2, 1)−N (X). In Theorem 19, the condition “µs ⊆ µv” where s, v = 1, 2 ; s ̸= v” is necessary as shown in Example 20. Y. Farhat, V. Subramanian / Eur. J. Pure Appl. Math, 16 (4) (2023), 2049-2065 2058 Example 20. Take X = {e, f, k, l}; µ1 = {∅, {e, k}, {e, l}, {f, l}, {e, f, l}, {e, k, l}, X} and µ2 = {∅, {e, f}, {f, k}, {e, l}, {f, l}, {e, f, k}, {e, f, l}, {e, k, l}, {f, k, l}, X}. Let P = {f, k}. Then i1(c1(P )) = i1({f, k}) = ∅ and so P is µ1-nowhere dense set. But P /∈ (2, 1) − N (X). Let M = {e, k}. Then i2(c2(M)) = i2({e, k}) = ∅ and so M is a µ2- nowhere dense set. ButM /∈ (1, 2)−N (X). Let C = {k, l}. Then i2(c2(C)) = i2({k, l}) = ∅ and so C is a µ2-nowhere dense set. But C /∈ (2, 1)−N (X). Consider the BGTS (X,µ1, µ2), X = [0, 3]; µ1 = {∅, [0, 32), (1, 2], [0, 2]} and µ2 = {∅, [0, 1), (1, 2), [0, 2)} Let D = [32 , 3]. Then D is a µ1-nowhere dense set in X. But D /∈ (1, 2)−N (X). µv − nowhere dense (s, v)− nowhere dense µs − nowhere dense The below Theorem 22 describes the above diagram. Example 21 proves the existence of the below Theorem 22. Example 21. (a) Fix s = 1, v = 2. Consider the bigeneralized topological space (X,µ1, µ2) where X = {p, q, r, s}; µ1 = {∅, {p, q}, {p, r}, {q, r}, {p, q, r}} and µ2 = {∅, {p, r}, {q, r}, {p, q, r}}. Obviously, µ2 ⊂ µ1. Consider, L = {q, s}. Then i1(c2(L)) = ∅ and so L ∈ (1, 2) −N (X). Here, i1(c1(L)) = ∅ and i2(c2(L)) = ∅. Thus, L is a µ1-nowhere dense set and also a µ2-nowhere dense set. (b) Fix s = 2, v = 1. Consider the bigeneralized topological space (X,µ1, µ2) where X = {p, q, r, s}; µ1 = {∅, {p, s}, {q, s}, {p, q, s}} Y. Farhat, V. Subramanian / Eur. J. Pure Appl. Math, 16 (4) (2023), 2049-2065 2059 and µ2 = {∅, {p}, {p, s}, {q, s}, {p, q, s}}. Clearly, µ1 ⊂ µ2. Take K = {q, r} then we get i2(c1(K)) = ∅ and hence K ∈ (2, 1)−N (X). Now, i1(c1(K)) = ∅ and i2(c2(K)) = ∅ which implies that K is a µ1-nowhere dense set and also a µ2-nowhere dense set. Theorem 22. Let µ1, µ2 be two generlized topologies on X and µv ⊆ µs where s, v = 1, 2 ; s ̸= v. If Q ∈ (s, v) − N (X), then Q is µv-nowhere dense and also µs-nowhere dense where s, v = 1, 2 ; s ̸= v. Proof. We give the detailed proof for s = 1 and v = 2 only. Assume that, µ2 ⊆ µ1. Let Q be a (1, 2)-nowhere dense set. Then i1(c2(Q)) = ∅. Suppose i1(c1(Q)) ̸= ∅. By assumption, i1(c2(Q)) ̸= ∅ which is a contradiction. Therefore, i1(c1(Q)) = ∅. If i2(c2(Q)) ̸= ∅, then there is a set M ∈ µ̃2 such that M ⊂ c2(Q). By assumption, M ∈ µ̃1. Thus, i1(c2(Q)) ̸= ∅ which is a contradiction. Therefore, i2(c2(Q)) = ∅. Theorem 23. Let (X,µ1, µ2) be a BGTS and K ⊂ X. If K ∈ (s, v)−N (X) then cv(K)− K ∈ (s, v)−N (X) where s, v = 1, 2 and s ̸= v. Proof. Let K ∈ (s, v) −N (X) where s, v = 1, 2 ; s ̸= v. Take s = 1 and v = 2. Then K is a (1, 2)-nowhere dense set in X. Since c2(K)−K ⊂ c2(K) we have c2(c2(K)−K) ⊂ c2(c2(K)). By Lemma 5 (e), c2(c2(K)−K) ⊂ c2(K). Then i1(c2(c2(K)−K)) ⊂ i1(c2(K)) and so i1(c2(c2(K)−K)) = ∅, by assumption. Therefore, c2(K)−K ∈ (1, 2)−N (X). By similar argument in the above case, we get c1(K)−K ∈ (2, 1)−N (X). Example 24. Consider the bigeneralized topological space (X,µ1, µ2), X = {e, f, k, l}; µ1 = {∅, {e, k}, {f, k}, {e, f, k}} and µ2 = {∅, {k}, {e, k}, {f, k}, {e, f, k}}. Take Q = {k} we get c2(Q) − Q = {e, f, l} and so i1(c2(c2(Q) − Q)) = ∅. Thus, c2(Q)−Q ∈ (1, 2)−N (X). But Q /∈ (1, 2)−N (X). Choose L = {f, k} so that c1(L) − L = {e, l} and so i2(c1(c1(L) − L)) = ∅ implies that c1(L)− L ∈ (2, 1)−N (X). But L /∈ (2, 1)−N (X). Theorem 25. Let (X,µ1, µ2) be a BGTS. For s, v = 1, 2 and s ̸= v, if D ∈ (s, v)−N (X), then the followings are true. (a) K ⊈ D for all K is a non-null (s, v)-µ-preopen set in X. (b) K ⊈ D for all K is a non-null (s, v)-µ-regular open set in X. Y. Farhat, V. Subramanian / Eur. J. Pure Appl. Math, 16 (4) (2023), 2049-2065 2060 (c) K ⊈ D for all K is a non-null (s, v)-open set in X. (d) K ⊈ D for all K is a non-null (s, v)-µ-α-open set in X. Proof. We give the detailed proof for (a) only. Assume that, D ∈ (s, v)−N (X) where s, v = 1, 2 and s ̸= v. Then is(cv(D)) = ∅ where s, v = 1, 2 and s ̸= v. Suppose there is a non-null (s, v)-µ-preopen set M in X such that M ⊂ D (7) where s, v = 1, 2 and s ̸= v. Here, M ⊂ is(cv(M)) (8) where s, v = 1, 2 and s ̸= v. From (7), we have is(cv(M)) ⊂ is(cv(D)) which implies that M ⊂ is(cv(D)) where s, v = 1, 2 and s ̸= v, by (8). Then is(cv(D)) ̸= ∅ which is not possible. Therefore, there is no non-null (s, v)-µ-preopen set M in X such that M ⊂ D where s, v = 1, 2 and s ̸= v. Hence D does not contain any non-null (s, v)-µ-preopen set in X where s, v = 1, 2 and s ̸= v. Theorem 26. Let (X,µ1, µ2) be a BGTS. If D ∈ (s, v) − N (X), then K ⊈ D for all K ∈ µ̃s where s, v = 1, 2 ; s ̸= v. Proof. Assume that, D ∈ (s, v)−N (X) where s, v = 1, 2 ; s ̸= v. Take s = 1 and v = 2. Then D ∈ (1, 2) − N (X). If there is H ∈ µ1 such that H ⊂ D, then i1(H) ⊂ D and so i1(H) ⊂ c2(D). This implies i1(i1(H)) ⊂ i1(c2(D)). By Lemma 5 (e), i1(H) ⊂ i1(c2(D)). By assumption, H ⊂ i1(c2(D)). Thus, i1(c2(D)) ̸= ∅ which is not possible. Therefore, D does not contain any non-null µ1-open set. Take s = 2 and v = 1. Then D ∈ (2, 1)−N (X). By similar arguments in the above case, we get the proof. In the rest of this section, we introduce a new tool namely, (s, v)-codense, and give some of its properties in a BGTS (X,µ1, µ2). Definition 27. Let (X,µ1, µ2) be a BGTS and E ⊂ X. Then E is (s, v)-codense if cs(cv(X − E)) = X where s, v = 1, 2 and s ̸= v. Example 28. Consider the bigeneralized topological space (X,µ1, µ2) whereX = {e, f, k, l}; µ1 = {∅, {e, f}, {f, l}, {e, f, l}} and µ2 = {∅, {e, k}, {f, k}, {e, f, k}}. Take A = {k, l} we get X − A = {e, f} and so c1(c2({e, f})) = X. Thus, A is a (1, 2)- codense set in X. Also, c2(c1({e, f})) = X. Therefore, A is a (2, 1)-codense set. Hence A is (s, v)-codense where s, v = 1, 2 and s ̸= v. Y. Farhat, V. Subramanian / Eur. J. Pure Appl. Math, 16 (4) (2023), 2049-2065 2061 Theorem 29. In a BGTS (X,µ1, µ2), if E ∈ (s, v)−N (X), then E is µs-codense where s, v = 1, 2 and s ̸= v. Proof. Given E ∈ (s, v) − N (X) for s, v = 1, 2 ; s ̸= v. Then is(cv(E)) = ∅ and so X− (is(cv(E))) = X where s, v = 1, 2 and s ̸= v. This implies cs(X− (cv(E))) = X where s, v = 1, 2 and s ̸= v which implies that cs(X − E) = X for s = 1, 2. Therefore, E is a µs-codense set in X for s = 1, 2. Example 30 explains that the reverse implication of Theorem 29 need not be true. Example 30. Consider the BGTS (X,µ1, µ2) where X = {e, f, k, l, r}; µ1 = {∅, {e, f}, {e, k}, {e, f, k}, {e, f, l}, {e, f, k, l}} and µ2 = {∅, {e, f}, {f, l}, {e, r}, {e, f, l}, {e, f, r}, {e, f, l, r}}. Choose P = {f, k, r}, then c2(X − P ) = X. But P /∈ (2, 1) − N (X). For, i2(c1(P )) = i2(X) = {e, f, l, r} ≠ ∅. Consider, Q = {f, l, r} we have c1(X −Q) = c1({e, k}) = X. But Q /∈ (1, 2)−N (X). For, i1(c2(Q)) = i1(X) = {e, f, k, l} ≠ ∅. Proposition 31. Let (X,µ1, µ2) be a BGTS. Then E is a (s, v)-codense set in X if and only if is(iv(E)) = ∅ where s, v = 1, 2 and s ̸= v. Proposition 32. Let (X,µ1, µ2) be a BGTS. If E ∈ (s, v) − N (X), then E is a (s, v)- codense set in X. Proposition 33. Let (X,µ1, µ2) be a BGTS. Then E ∈ (s, v)−D(X) if and only if X−E is (s, v)-codense where s, v = 1, 2 and s ̸= v. Proposition 34. Let (X,µ1, µ2) be a BGTS. If E is a (s, v)-codense set in X, then there is no non-null (s, v)-open set H such that H ⊂ E where s, v = 1, 2 and s ̸= v. The reverse implication of Proposition 34 is generally not true as given by the below Example 35. Example 35. (a) Consider the bigeneralized topological space (X,µ1, µ2) where X = [0, 4]; µ1 = {∅, [0, 2), (1, 3], [0, 3]} and µ2 = {∅, [0, 32), (1, 3], [0, 3]}. Let A = [0, 2). Here B = (1, 3] is (2, 1)-open set. Also, B ⊈ A. But i2(i1(A)) = [0, 32) ̸= ∅. (b) Consider the bigeneralized topological space (X,µ1, µ2) where X = [0, 3]; Y. Farhat, V. Subramanian / Eur. J. Pure Appl. Math, 16 (4) (2023), 2049-2065 2062 µ1 = {∅, [0, 32), (1, 2), (1, 3), [0, 3)} and µ2 = {∅, [0, 2), (1, 3), [0, 3)}. Take A = [0, 2). Here B = (1, 3) is (1, 2)-open set. Also, B ⊈ A. But i1(i2(A)) = [0, 32) ̸= ∅. 5. Sets via Functions In this section, we give some properties for (s, v)-dense and (s, v)-nowhere dense sets under generalized continuous functions in a bigeneralized topological space. Now, we recall some basic definitions defined in [4]. Let (X,µ1 X , µ2 X) and (Y, µ1 Y , µ 2 Y ) be two BGTS and h : (X,µ1 X , µ2 X) → (Y, µ1 Y , µ 2 Y ) be a map. Then • h is called (s, v)-generalized continuous (µ(s,v)-continuous) if h−1(B) is µ(s,v)-closed in X for every µv-closed B of Y where s, v = 1, 2 and s ̸= v. • h is called as µs-continuous if h−1(C) is µs-closed in X for every µs-closed C of Y for s = 1, 2. • h is said to be µs-open if h(D) is µs-open of Y for every µs-open D of X for s = 1, 2. Theorem 36. Let (X,µ1 X , µ2 X) and (Y, µ1 Y , µ 2 Y ) be two bigeneralized topological spaces, h : (X,µ1 X , µ2 X) → (Y, µ1 Y , µ 2 Y ) be a µ(s,v)-continuous function where s, v = 1, 2 and s ̸= v. If Q∩P ̸= ∅ for every P is non-null µ(s,v)-open, then h(Q) ∈ (s, v)−D(Y ) where Q ⊂ X; s, v = 1, 2 and s ̸= v. Proof. It is enough to prove, h(Q) ∈ D(µv Y ) where v = 1, 2, by Theorem 4. Take v = 2. Let P ∈ µ̃2 Y . Then Y − P is µ2 Y -closed. By hypothesis, h−1(Y − P ) is µ(1,2)-closed in X. Then h−1(P ) is non-null µ(1,2)-open. By hypothesis, Q ∩ h−1(P ) ̸= ∅. This implies h−1(h(Q)) ∩ h−1(P ) ̸= ∅ which implies that h−1(h(Q) ∩ P ) ̸= ∅. Thus, h(Q) ∩ P ̸= ∅. Hence h(Q) ∈ D(µ2 Y ). Take v = 1. Then by the same arguments in the above case, we get h(Q) ∈ D(µ1 Y ). Hence h(Q) ∈ D(µv Y ) where v = 1, 2. Theorem 37. Let (X,µ1 X , µ2 X) and (Y, µ1 Y , µ 2 Y ) be two bigeneralized topological spaces, P,Q ⊂ X, h : (X,µ1 X , µ2 X) → (Y, µ1 Y , µ 2 Y ) be a µs-continuous function for s = 1, 2. Then the followings are true. (a) If P ∈ D(µv), then h(P ) ∈ (s, v)−D(Y ) where s, v = 1, 2 and s ̸= v. (b) If Q is µv-codense and h is one-one, then h(Q) is (s, v)-codense in Y where s, v = 1, 2 and s ̸= v. Proof. (a). It is enough to prove, h(P ) ∈ D(µv) in Y where v = 1, 2, by Theorem 4. Assume that, P ∈ D(µv) in X for v = 1, 2. Take v = 1. Then P ∈ D(µ1) in X. Let M ∈ µ̃1. Then Y −M is µ1-closed in Y. By hypothesis, h−1(Y −M) is µ1-closed set in X. Then h−1(M) is a non-null µ1-open set in X. By hypothesis, P ∩h−1(M) ̸= ∅. This implies Y. Farhat, V. Subramanian / Eur. J. Pure Appl. Math, 16 (4) (2023), 2049-2065 2063 h−1(h(P )) ∩ h−1(M) ̸= ∅, since P ⊂ h−1(h(P )) which implies that h−1(h(P ) ∩M) ̸= ∅. Thus, h(P ) ∩M ̸= ∅. Hence h(P ) ∈ D(µ1) in Y. Take v = 2. Then by similar arguments in the above case, we get h(P ) ∈ D(µ2) in Y. Hence h(P ) ∈ D(µv) in Y where v = 1, 2. (b) Let Q be a µv-codense set in X for v = 1, 2. Then X −Q ∈ D(µv) in X for v = 1, 2. By (a), h(X − Q) ∈ (s, v) − D(Y ) where s, v = 1, 2 and s ̸= v. Since h is one-one, h(X)−h(Q) ∈ (s, v)−D(Y ) where s, v = 1, 2 and s ̸= v. Therefore, Y−h(Q) ∈ (s, v)−D(Y ) where s, v = 1, 2 and s ̸= v. Hence h(Q) is (s, v)-codense in Y where s, v = 1, 2 ; s ̸= v. Theorem 38. Let (X,µ1 X , µ2 X) and (Y, µ1 Y , µ 2 Y ) be two bigeneralized topological spaces, K,L ⊂ Y, h : (X,µ1 X , µ2 X) → (Y, µ1 Y , µ 2 Y ) be a µs-open, one-one function for s = 1, 2. Then the followings are true. (a) If K ∈ D(µv) in Y, then h−1(K) ∈ (s, v)−D(X) where s, v = 1, 2 and s ̸= v. (b) If L is µv-codense in Y, then h−1(L) is (s, v)-codense where s, v = 1, 2 and s ̸= v. Proof. The trivial proof is omitted. 6. (s, v)-dense sets applications In 1999, Molodstov introduced a new mathematical tool namely, soft set theory [11]. It has been used for dealing with uncertainty. Most of the researchers presented an application of soft sets in decision-making problems. Motivated, by this we try to give an example of the soft set using (s, v)-dense and (s, v)-nowhere dense sets in a bigeneralized topological space. Example 39. Consider the BGTS (X,µ1, µ2) where X = {a, b, c, d}; µ1 = {∅, {a, b}, {a, c}, {a, d}, {a, b, c}, {a, b, d}, {a, c, d}, X}; and µ2 = {∅, {b, c}, {b, d}, {b, c, d}}. Here, • (1, 2)−D(X) = exp(X) where exp(X) is the power set of X. • (2, 1)−D(X) = {{a}, {b}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}, X}. Let U = {a, c} be a subset of X and E = {(1, 2)-dense set, (2, 1)-dense set, both} = {e1, e2, e3} is the set of parameters. Define a map F from E to exp(U) by, F (e1) = {c};F (e2) = {a};F (e3) = {a, c}. Then the pair (F,E) is a soft set over U. Example 40. Consider the bigeneralized topological space (X,µ1, µ2) whereX = {a, b, c, d}; µ1 = {∅, {b}, {a, b}, {a, c}, {a, b, c}} Y. Farhat, V. Subramanian / Eur. J. Pure Appl. Math, 16 (4) (2023), 2049-2065 2064 and µ2 = {∅, {a}, {a, c}, {b, c}, {a, b, c}}. Here, • (1, 2)−N (X) = {∅, {a}, {d}, {a, d}}; • (2, 1)−N (X) = {∅, {b}, {c}, {d}, {b, d}, {c, d}}. Let U = {a, c, d} be a subset of X and E = {(1, 2)-nowhere dense set, (2, 1)-nowhere dense set, both} = {e1, e2, e3} is the set of parameters. Consider the map F from E into the power set of U. Defined by F (e1) = {a};F (e2) = {c};F (e3) = {d}. Then (F,E) is a soft set over U. Example 41. Consider the bigeneralized topological space (X,µ1, µ2) whereX = {a, b, c, d}; µ1 = {∅, {a}, {a, d}, {c, d}, {a, c, d}} and µ2 = {∅, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}. Here, • (1, 2)-codense sets = {∅, {a}, {b}, {c}, {d}, {a, d}, {b, c}, {b, d}, {c, d}, {b, c, d}}. • (2, 1)-codense sets = {∅, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {a, b, c}, {a, b, d}}. Let U = {a, c, d} be a subset of X and E = {(1, 2)-codense set, (2, 1)-codense set, (1, 2)- codense but not (2, 1)-codense, (2, 1)-codense but not (1, 2)-codense, (1, 2)-codense and (2, 1)-codense } = {e1, e2, e3, e4, e5} is the set of parameters. Consider the map F from E into the power set of U. Defined by F (e1) = {a};F (e2) = {c};F (e3) = {c, d};F (e4) = {a, c};F (e5) = {d}. Then we get the pair (F,E) is a soft set over U. Example 42. Consider the generalized topological space (X, η1, η2) whereX = {a, b, c, d}; η1 and η2 are defined in above Example 40, that is; we take η1 = µ2 and η2 = µ1. Then we get; • η1-nowhere dense sets = {∅, {b}, {d}, {b, d}}; • η2-nowhere dense sets = {∅, {c}, {d}, {c, d}}; • η1-dense sets = {{a, b}, {a, c}, {a, b, c}, {a, b, d}, {a, c, d}, X}; • η2-dense sets = {{a, b}, {b, c}, {a, b, c}, {a, b, d}, {b, c, d}, X}; • (1, 2)−N (X) = {∅, {b}, {c}, {d}, {b, d}, {c, d}}; • (2, 1)−N (X) = {∅, {a}, {d}, {a, d}}. Let U = {a, b, c} be a non-null subset of X and E = {η1-nowhere dense set, η2-nowhere dense set, η1-dense set, η2-dense set, (1, 2)-nowhere dense set, (2, 1)-nowhere dense set} = {e1, e2, e3, e4, e5, e6} is the set of parameters. Take F be a function defined from E to the subsets of U by; F (e1) = {b};F (e2) = {c};F (e3) = {a, c};F (e4) = {b, c};F (e5) = {d};F (e6) = {a}. Thus, (F,E) is a soft set over U. REFERENCES 2065 7. Conclusion In this article, various properties for (s, v)-dense and (s, v)-nowhere dense sets are proved, which are useful to easily check the characterization of a given set in a bigeneralized topological space. References [1] Chawalit Boonpok. Weakly open functions on bigeneralized topological spaces. Int. Journal of Math. Analysis, 4(18):891–897, 2010. [2] Akos Császár. Generalized open sets. Acta mathematica hungarica, 75, 1997. [3] Akos Császár. Generalized open sets in generalized topologies. Acta mathematica hungarica, 106, 2005. 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