EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 4, 2023, 2286-2305 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Dense Sets Diaa Elgezouli1, Mutaz Omer2, Yasser Farhat3,∗, Elmhadi Afif4, Vadakasi Subramanian5 1 Department of Basic Sciences, King Saud University, Saudi Arabia 2 University of Khartoum, Khartoum, Sudan 3 Academic Support Department, Abu Dhabi Polytechnic, P. O. Box 111499, Abu Dhabi, UAE. 4 University of Gezira, Sudan 5 Department of Mathematics, A.K.D.Dharma Raja Women’s College, Rajapalayam., India Abstract. In this paper, we introduce one interesting mathematical tool namely, (s, v)⋆-dense, and analyze its nature in a bigeneralized topological space. Further, we prove some properties of this set and give the relationship between (s, v)-dense and (s, v)⋆-dense sets. Finally, we give applications for various sets defined in a bigeneralized topological space. 2020 Mathematics Subject Classifications: AMS Subject Classification(2020):Primary: 54A05, 54A10. Key Words and Phrases: Bigeneralized topological space, µ(s,v)-open, (s, v)-open, (s, v)-nowhere dense. 1. Introduction The concept of a generalized topological space was introduced by Császár in [3]. Some researchers have defined various concepts in this space and examined their signif- icance in a generalized topological space. Especially, in a generalized topological space, dense sets were introduced by Ekici [8]. He has proven few results for dense sets in a generalized topological space. Based on this, some mathematicians have proved various properties for dense sets e.g. [11, 12, 15, 17, 19]. In [10], J.C. Kelly introduced the concept namely, a bitopological space. Using these aspects, Boonpok founded the notion of a bigeneralized topological space in 2010 [2]. He examines the significance of (m,n)-closed sets in a bigeneralized topological space. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i4.4944 Email addresses: dbushra.c@ksu.edu.sa (D. Elgezouli), farhat.yasser.1@gmail.com (Y. Farhat) https://www.ejpam.com 2286 © 2023 EJPAM All rights reserved. D. Elgezouli et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2286-2305 2287 Inspired by all this, we define a new dense set, namely, (s, v)⋆-dense set using semi-open sets in a bigeneralized topological space. Find various interesting results for (s, v)⋆-dense sets. Next section, the preliminary definitions, and lemmas are remembered. In sections 3 & 4, in a bigenerlized topological space, examined the significance of (s, v)⋆-dense set. The relationship between µ-dense and (s, v)⋆-dense sets are proven. Further, few results for (s, v)⋆-dense sets using functions are launched. In the last section, we defined a soft set using (s, v)⋆-dense sets and various types of open sets defined in a bigeneralized topological space. 2. Preliminaries In [3], let X be any non-null set. A family µ of subsets of X is a generalized topology in X if it contains the empty set and is closed under arbitrary union. The pair (X,µ) is called a generalized topological space (GTS). If X ∈ µ, then (X,µ) is called a strong generalized topological space (sGTS). In [6], if Q ∈ µ, then Q is called a µ-open set and if X −Q ∈ µ, then Q is said to be a µ-closed set. The interior of Q ⊂ X denoted by iµ(D), is the union of all µ-open sets contained in D and the closure of D denoted by cµ(D), is the intersection of all µ-closed sets containing D [12]. Here, the interior and closure of the set Q are notated by iQ and cQ, respectively, when no confusion can arise. In [11], notated by; µ̃ = {D ∈ µ | D ̸= ∅}; µ(x) = {D ∈ µ | x ∈ D}. Definition 1. [8] A subset Q of a GTS (X,µ) is said to be; • µ-nowhere dense if icQ = ∅. • µ-dense if cQ = X. • µ-codense [7] if c(X −Q) = X. Definition 2. [11] A subset Q of X is called as; • µ-meager if Q = ⋃ m∈NQm where each Qm is a µ-nowhere dense set. • µ-second category if Q is not µ-meager. In [11], defined two new generalized topologies; µ⋆ = { ⋃ t(L t 1 ∩ Lt 2 ∩ Lt 3 ∩ ... ∩ Lt nt ) | Lt 1, L t 2, ..., L t nt ∈ µ}; µ⋆⋆ = {D ⊂ X | D is of µ-II category}. D. Elgezouli et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2286-2305 2288 Obviously, µ ⊂ µ⋆ and µ⋆ is closed under finite intersection [11]. Definition 3. [6] Let (X,µ) be a GTS and Q ⊂ X is called; • µ-semi-open if Q ⊂ cµ(iµ(Q)). • µ-pre-open if Q ⊂ iµ(cµ(Q)). • µ-α-open if Q ⊂ iµ(cµ(iµ(Q))). • µ-β-open if Q ⊂ cµ(iµ(cµ(Q))). • µ-b-open [1] if Q ⊂ cµ(iµ(Q)) ∪ iµ(cµ(Q)). Moreover, σ(µ) or σ(µ(X)) = {Q ⊂ X | Q is µ-semi-open set in X} [12]. The µ- semi-interior of a subset Q of (X,µ), denoted by iσ(Q), is defined by the union of all µ-semi-open subsets of X contained in Q [12]. Definition 4. [2] Let µ1 and µ2 be two generalized topologies defined a non-null set X. A triple (X,µ1, µ2) is called a bigeneralized topological space (briefly, BGTS). • The closure and interior of Q ⊂ X with respect to µs are denoted by cs(Q) and is(Q), respectively, for s = 1, 2. • Q is called (s, v)-closed if cs(cv(Q)) = D, where s, v = 1 or 2 ; s ̸= v. • Q is called (s, v)-open if X −Q is (s, v)-closed where s, v = 1 or 2 ; s ̸= v. A subset Q of a BGTS (X,µ1, µ2) is said to be (1) (s, v)-µ-regular open if Q = is(cv(Q)) where s, v = 1 or 2 ; s ̸= v. (2) (s, v)-µ-semi-open if Q ⊆ cv(is(Q)) where s, v = 1 or 2 ; s ̸= v. (3) (s, v)-µ-preopen if Q ⊆ is(cv(Q)) where s, v = 1 or 2 ; s ̸= v. (4) (s, v)-µ-α-open if Q ⊆ is(cv(is(Q))) where s, v = 1 or 2 ; s ̸= v [2]. Lemma 1. [2, Proposition 3.4] Let (X,µ1, µ2) be a BGTS and Q ⊂ X. Then Q is (s, v)- closed if and only if Q is both µ-closed in (X,µs) and (X,µv) where s, v = 1, 2 ; s ̸= v. Lemma 2. [5] In a GTS (X,µ), r ∈ cP if and only if L ∩ P ̸= ∅ for all L ∈ µ̃(r). Lemma 3. [12, Lemma 3.2] Let (X,µ) be a GTS and K,P ⊂ X. If K ∈ µ̃ and K∩P = ∅, then K ∩ cP = ∅. Lemma 4. [13, Proposition 2.2] Let (X,µ) be a GTS. For subsets Q,P ⊂ X, then the following properties holds: (a) cµ(X −Q) = X − iµ(Q) and iµ(X −Q) = X − cµ(Q). (b) If X −Q ∈ µ, then cµ(Q) = Q and if Q ∈ µ, then iµ(Q) = Q. (c) If Q ⊆ P, then cµ(Q) ⊆ cµ(P ) and iµ(Q) ⊆ iµ(P ). (d) Q ⊆ cµ(Q) and iµ(Q) ⊆ Q. (e) cµ(cµ(Q)) = cµ(Q) and iµ(iµ(Q)) = iµ(Q). 3. Nature of (s, v)⋆-dense sets Here, we define another branch of dense set namely, (s, v)⋆-dense set and study its significance in a BGTS. D. Elgezouli et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2286-2305 2289 In a bigeneralized topological space, various interesting results for (s, v)⋆-dense sets are derived which is helpful for examining the given set is (s, v)⋆-dense or not. Definition 5. A GTS (X,µ) is called as; • hyperconnected [8] if cµ(Q) = X whenever Q ∈ µ̃. • generalized submaximal [7] if Q ∈ µ̃ whenever cµ(Q) = X. Definition 6. [16] A GT µ onX is said to satisfy the I-property wheneverW1,W2, ..,Wm ∈ µ with W1 ∩W2 ∩ · · · ∩Wm ̸= ∅, iµ(W1 ∩W2 ∩ · · · ∩Wm) ̸= ∅. Definition 7. [9] A non-null subset Q of a BGTS (X,µ1, µ2) is called (s, v)-dense if cs(cv(Q)) = X where s, v = 1, 2 and s ̸= v. Moreover, (s, v) − D(X) = {Q ⊂ X | Q is a (s, v)-dense set in X} where s, v = 1, 2 ; s ̸= v. Definition 8. Let Q be a non-null subset of a bigeneralized topological space (X,µ1, µ2). Then Q is called (µs, µv) ⋆-dense (briefly, (s, v)⋆-dense) if cv(Q)∩M ̸= ∅ for every M ∈ σ̃s where s, v = 1, 2 ; s ̸= v;σs = σ(µs). For simplification we noted; (s, v)⋆ −D(X) = {Q ⊂ X | Q is a (s, v)⋆-dense set in X} where s, v = 1, 2 ; s ̸= v. Remark 9. In a BGTS, if P ∈ (s, v)⋆ −D(X) and P ⊂ Q, then Q(s, v)⋆ −D(X). Example 10. Consider the bigeneralized topological space (X,µ1, µ2) whereX = {p, q, r, s}; µ1 = {∅, {p, q}, {q, r}, {p, q, r}} and µ2 = {∅, {p, s}, {q, s}, {p, q, s}}. Then σ1 = {∅, {p, q}, {q, r}, {p, q, r}, {p, q, s}, {q, r, s}, X}. Take K = {q, r}. Then c2(K) = K. Also, K ∩M ̸= ∅ for all M ∈ σ̃1. Thus, c2(K)∩M ̸= ∅ for all M ∈ σ̃1. Therefore, K ∈ (1, 2)⋆ −D(X). (b) Consider the bigeneralized topological space (X,µ1, µ2) where X = {p, q, r, s}; µ1 = {∅, {q, r}, {q, s}, {q, r, s}} and µ2 = {∅, {p, q}, {p, r}, {p, q, r}}. D. Elgezouli et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2286-2305 2290 Then σ2 = {∅, {p, q}, {p, r}, {p, q, r}, {p, q, s}, {p, r, s}, X}. Take J = {p, r}. Here c1(J) ∩H ̸= ∅ for all H ∈ σ̃2. Hence J ∈ (2, 1)⋆ −D(X). Theorem 11. Let (X,µ1, µ2) be a BGTS and cµs(Q) = X. If µs is a sGT, then Q ∈ (s, v)⋆ −D(X) where s, v = 1, 2 ; s ̸= v. Proof. Take s = 1 and v = 2. Assume that, cµ1(Q) = X and µ1 is a sGT. Let P ∈ σ̃1. Then P ⊂ cµ1(iµ1(P )) and so iµ1(P ) ̸= ∅, since µ1 is a sGT. This implies iµ1(P ) ∈ µ̃1 which implies that iµ1(P ) ∩Q ̸= ∅. Thus, Q ∩ P ̸= ∅. Therefore, Q ∈ (1, 2)⋆ −D(X). Take s = 2 and v = 1. Suppose cµ2(Q) = X and µ2 is a sGT. Let M ∈ σ̃2. Then M ⊂ cµ2(iµ2(M)) and so iµ2(M) ̸= ∅, since µ2 is a sGT. Thus, iµ2(M) ∈ µ̃2 so that iµ2(M) ∩Q ̸= ∅. This implies Q ∩M ̸= ∅ which implies that Q ∈ (2, 1)⋆ −D(X). The below Example 12 shows that the hypothesis in Theorem 11 can not be dropped. Example 12. (a). Consider the BGTS (X,µ1, µ2) where X = {p, q, r, s}; µ1 = {∅, {q, s}, {r, s}, {q, r, s}} and µ2 = {∅, {p, r}, {q, r}, {p, q, r}, {p, q, s}, X}. Fix s = 1; v = 2. Obviously, σ1 = {∅, {p}, {q, s}, {r, s}, {p, q, s}, {p, r, s}, {q, r, s}, X}. Choose L = {q, s} so that cµ1L = X and cµ2L = L. Thus, L = {q, s} is µ1-dense. But cµ2L ∩ {p} = ∅ where {p} ∈ σ̃1 for that L /∈ (1, 2)⋆ −D(X). (b). Consider the BGTS (X,µ1, µ2) where X = {p, q, r, s}; µ1 = {∅, {p, r}, {q, r}, {r, s}, {p, q, r}, {p, r, s}, {q, r, s}, X} and µ2 = {∅, {p, s}, {q, s}, {p, q, s}}. Fix s = 2; v = 1. Obviously, σ2 = {∅, {r}, {p, s}, {q, s}, {p, q, s}, {p, r, s}, {q, r, s}, X}. Choose K = {s} so that cµ1K = K and cµ2K = X. Here, K = {s} is µ2-dense. But cµ1K ∩ {r} = ∅ where {r} ∈ σ̃2 for that K /∈ (2, 1)⋆ −D(X). D. Elgezouli et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2286-2305 2291 Theorem 13. Let (X,µ1, µ2) be a BGTS. Then the following are true. (a) If cµv(Q) = X, then Q ∈ (s, v)⋆ −D(X) where s, v = 1, 2 ; s ̸= v. (b) If µs ⊂ µv, then every (s, v)⋆-dense is µs-dense where s, v = 1, 2 ; s ̸= v. Proof. (a). Assume that, cµv(Q) = X for v = 1, 2. Fix s = 1 and v = 2. We get cµ2(Q) = X so that cµ2(Q)∩H ̸= ∅ for all H ∈ σ̃1. Therefore, Q is (1, 2)⋆ −D(X). Take s = 2 and v = 1. Then cµ1(Q) = X and so cµ1(Q)∩K ̸= ∅ for all K ∈ σ̃2. Therefore, Q is (2, 1)⋆ −D(X). (b). Suppose that µs ⊂ µv for s, v = 1, 2 ; s ̸= v. Let K ∈ (s, v)⋆ −D(X) where s, v = 1, 2 ; s ̸= v. Consider s = 1 and v = 2. Then µ1 ⊂ µ2 and K ∈ (1, 2)⋆ − D(X). Let G ∈ µ̃1. Then G ∈ σ̃1 so that G∩cµ2K ̸= ∅. By hypothesis and Lemma 3, G∩K ̸= ∅. HenceK is µ1-dense. Take s = 2 and v = 1. Then µ2 ⊂ µ1 and K ∈ (2, 1)⋆ −D(X). Let H ∈ µ̃2. Then H ∈ σ̃2 so that H ∩ cµ1K ̸= ∅. By hypothesis and Lemma 3, H ∩K ̸= ∅. Hence K is µ2-dense. The below Example 14 (b) shows that the converse part of Theorem 13 (a) need not be true and the hypothesis of Theorem 13 (b) can not be neglected as shown by Example 14 (a). Example 14. Consider the bigeneralized topological space (X,µ1, µ2) whereX = {p, q, r, s}; µ1 = {∅, {p, r}, {p, s}, {p, r, s}} and µ2 = {∅, {q, r}, {q, s}, {r, s}, {q, r, s}}. We get σ1 = {∅, {q}, {p, r}, {p, s}, {p, q, r}, {p, q, s}, {p, r, s}, X} and σ2 = {∅, {p}, {q, r}, {q, s}, {r, s}, {p, q, r}, {p, q, s}, {p, r, s}, {q, r, s}, X}. (a). Fix s = 1; v = 2. Here, µ1 ⊈ µ2. Choose Q = {q, r} we get Q ∈ (1, 2)⋆ − D(X). Because, c2Q ∩ L ̸= ∅ for all L ∈ σ̃1. But c1Q = Q ̸= X so that Q is not µ1-dense. Take s = 2, v = 1 and L = {p, q}. Here, c1L ∩ D ̸= ∅ for each D ∈ σ̃2 so that L ∈ (2, 1)⋆ −D(X). Since c2L = L ̸= ∅ we have L is not µ2-dense. D. Elgezouli et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2286-2305 2292 (b). Fix s = 1; v = 2. Choose W = {p, q} we get W ∈ (1, 2)⋆ −D(X), since c2W ∩ L ̸= ∅ for all L ∈ σ̃1. Here, c2W = W ̸= X so that W is not a µ2-dense set. Take s = 2; v = 1. Consider the BGTS (X,µ1, µ2) where X = {p, q, r, s}; µ1 = {∅, {p, r}, {p, s}, {r, s}, {p, r, s}} and µ2 = {∅, {q, r}, {q, s}, {q, r, s}}. Clearly, we have σ1 = {∅, {q}, {p, r}, {p, s}, {r, s}, {p, q, r}, {p, q, s}, {p, r, s}, {q, r, s}, X} and σ2 = {∅, {p}, {q, r}, {q, s}, {p, q, r}, {p, q, s}, {q, r, s}, X}. Consider K = {p, q}. Since c1K ∩M ̸= ∅ for all M ∈ σ̃2 we get K ∈ (2, 1)⋆ −D(X). But c1K = K ̸= X. Thus, K is not µ1-dense. Theorem 15. Let (X,µ1, µ2) be a BGTS. Then (s, v)⋆ − D(X) ⊂ (s, v) − D(X) where s, v = 1, 2 ; s ̸= v. Proof. Let Q ∈ (s, v)⋆ −D(X). Take s = 1; v = 2. Then Q ∈ (1, 2)⋆ − D(X) so that c2(Q) ∩ M ̸= ∅ for every M ∈ σ̃1. Since µ1 ⊂ σ1, c2(Q) ∩K ̸= ∅ for every K ∈ µ̃1. Therefore, Q ∈ (1, 2)−D(X). Fix s = 2; v = 1. We get Q ∈ (2, 1)⋆ − D(X) for that c1(Q) ∩ L ̸= ∅ for every L ∈ σ̃2. Since µ2 ⊂ σ2, c1(Q) ∩G ̸= ∅ for every G ∈ µ̃2. Therefore, Q ∈ (2, 1)−D(X). The below Example 16 shows that in a bigeneralized topological space, the reverse implication of the above Theorem 15 need not be true in general. Example 16. Consider the bigeneralized topological space (X,µ1, µ2) where X = {p, q, r, s}; µ1 = {∅, {p, r}, {r, s}, {p, r, s}} and µ2 = {∅, {p, q}, {p, r}, {q, r}, {p, q, r}}. Then σ1 = {∅, {q}, {p, r}, {r, s}, {p, q, r}, {p, r, s}, {q, r, s}, X} and D. Elgezouli et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2286-2305 2293 σ2 = {∅, {s}, {p, q}, {p, r}, {q, r}, {p, q, r}, {p, q, s}, {p, r, s}, {q, r, s}, X}. • Fix s = 1 and v = 2. Choose K = {p, s} we get K is (1, 2)-dense. But K /∈ (1, 2)⋆−D(X). For, if we choose D = {q}, then D ∈ σ̃1. But D ∩ c2(K) = ∅. Thus, there is D ∈ σ̃1 such that D ∩ c2(K) = ∅. • Fix s = 2 and v = 1. Take L = {p, q}, then we get L ∈ (1, 2) − D(X). Here, we take H = {s} so that H ∈ σ̃2 but H ∩ c1(L) = ∅. Thus, L /∈ (1, 2)⋆ −D(X). Theorem 17. Let (X,µ1, µ2) be a BGTS. If µs is a sGT, then (s, v)−D(X) ⊂ (s, v)⋆ − D(X) where s, v = 1, 2 ; s ̸= v. Proof. Assume that, µs is sGT and Q ∈ (s, v)−D(X). Take s = 1; v = 2. Then Q ∈ (1, 2)−D(X) so that c1(c2(Q)) = X. Thus, c2(Q) ∩M ̸= ∅ for all M ∈ µ̃1. Let H ∈ σ̃1. Suppose H ∈ µ̃1. Then there is nothing to prove. Suppose H /∈ µ̃1. Here H ⊂ c1(i1(H)). This implies c1(i1(H)) ̸= ∅ which implies that i1(H) ̸= ∅, by hypothesis. Thus, i1(H) ∈ µ̃1 so that c2(Q)∩H ̸= ∅. Thus, c2(Q)∩H ̸= ∅ for all H ∈ σ̃1. Hence Q ∈ (1, 2)⋆ −D(X). Fix s = 2; v = 1. We get Q ∈ (2, 1) − D(X) such that c2(c1(Q)) = X which implies c1(Q) ∩ L ̸= ∅ for all L ∈ µ̃2. Let K ∈ σ̃2. Suppose K ∈ µ̃2. Then there is nothing to prove. If K /∈ µ̃2, then from the definition of K such that K ⊂ c2(i2(K)). This implies c2(i2(K)) ̸= ∅ which implies that i2(K) ̸= ∅ since µ2 is a sGT. Thus, i2(K) ∈ µ̃2 so that c1(Q) ∩K ̸= ∅. Thus, c1(Q) ∩K ̸= ∅ for all K ∈ σ̃2. Hence Q ∈ (2, 1)⋆ −D(X). The above Example 16 also proves that the hypothesis of Theorem 17 can not be dropped. Theorem 18. Let (X,µ1, µ2) be a BGTS and µ1 ⊂ µ2. If µ1 ⊂ (s, v)⋆ − D(X), then (X,µ1) is hyperconnected for s, v = 1, 2 ; s ̸= v. Proof. Let Q ∈ µ̃1. Choose s = 1 and v = 2. Then Q ∈ (1, 2)⋆ − D(X). By hypothesis and Theorem 13, Q is µ1-dense so that (X,µ1) is a hyperconnected space. Fix s = 2 ; v = 1. Then Q ∈ (2, 1)⋆ −D(X) so that c1(Q) ∩M ̸= ∅ for every M ∈ σ̃2. Let K ∈ µ̃1. By hypothesis, K ∈ µ̃2 which implies K ∈ σ̃2, since µ2 ⊂ σ2 which turn implies that c1(Q)∩K ̸= ∅. Thus, Q∩K ̸= ∅. Since K is an arbitrary non-null µ1-open set, Q is µ1-dense. Therefore, (X,µ1) is a hyperconnected space. Theorem 19. Let (X,µ1, µ2) be a BGTS. If µ2 ⊂ µ1 and if µ2 ⊂ (s, v)⋆ − D(X) where s, v = 1, 2 ; s ̸= v, then (X,µ2) is hyperconnected. D. Elgezouli et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2286-2305 2294 Proof. Let P ∈ µ̃2. Take s = 1 and v = 2. Then P ∈ (1, 2)⋆ −D(X) so that c2(P )∩M ̸= ∅ for every M ∈ σ̃1. Let K ∈ µ̃2. By hypothesis, K ∈ µ̃1 which implies K ∈ σ̃1, since µ1 ⊂ σ1 which turn implies that c2(P ) ∩K ̸= ∅. Thus, P ∩K ̸= ∅. Since K is an arbitrary non-null µ2-open set, P is µ2-dense. Hence (X,µ2) is hyperconnected. Now we choose s = 2 ; v = 1. We get P ∈ (2, 1)⋆−D(X). By Theorem 13 and hypothesis, we get Q is µ2-dense. Therefore, (X,µ2) is a hyperconnected space. Theorem 20. Let (X,µ1, µ2) be a BGTS and Q ∈ (s, v) − D(X); Q ∈ µv; J ∈ (v, s)⋆ − D(X). If µv ⊂ µs and if µv has the I-property, then Q∩J ∈ (v, s)−D(X) where s, v = 1, 2 ; s ̸= v. Proof. Fix s = 1, v = 2. Assume that, Q ∈ (1, 2) − D(X); Q ∈ µ2 and J ∈ (2, 1)⋆ − D(X). Then (a) c1(c2(Q)) = X. (b) c1J ∩M ̸= ∅ for all M ∈ σ̃2. Suppose µ2 has the I-property and µ2 ⊂ µ1. Let K ∈ µ̃2. By hypothesis, K ∈ µ̃1 so that K ∩ c2(Q) ̸= ∅, by (a) which implies that K ∩ Q ̸= ∅, by Lemma 3. By our assumption, i2(K∩Q) ̸= ∅. By (b), c1J∩i2(K∩Q) ̸= ∅ which implies c2J∩i2(K∩Q) ̸= ∅ by hypothesis which turn implies that J ∩ i2(K ∩Q) ̸= ∅, by Lemma 3. Thus, J ∩ (K ∩Q) ̸= ∅ so that (J ∩Q) ∩K ̸= ∅. Therefore, c1(J ∩Q) ∩K ̸= ∅. Hence Q ∩ J ∈ (2, 1)−D(X). Take s = 2, v = 1. Assume that, Q ∈ (2, 1) − D(X); Q ∈ µ1 and J ∈ (1, 2)⋆ − D(X). We get (c) c2(c1(Q)) = X. (d) c2J ∩H ̸= ∅ for all H ∈ σ̃1. Suppose µ1 has the I-property and µ1 ⊂ µ2. Let L ∈ µ̃1. By hypothesis, L ∈ µ̃2 so that L ∩ c1(Q) ̸= ∅, by (c) which implies that L ∩ Q ̸= ∅, by Lemma 3. By our assumption, i1(L∩Q) ̸= ∅. By (d), c2J ∩ i1(L∩Q) ̸= ∅. This implies c1J ∩ i1(L∩Q) ̸= ∅ by hypothesis which implies that J ∩ i1(L ∩ Q) ̸= ∅, by Lemma 3. Thus, J ∩ (L ∩ Q) ̸= ∅ so that (J ∩Q) ∩ L ̸= ∅. Therefore, c2(J ∩Q) ∩ L ̸= ∅. Hence Q ∩ J ∈ (1, 2)−D(X). Moreover, in a BGTS every µv-dense set is (s, v)-preopen where s, v = 1, 2 ; s ̸= v. Theorem 21. Let (X,µ1, µ2) be a BGTS and η1 = {Q ⊂ X | Q ∈ (1, 2)⋆ − D(X); η2 = {P ⊂ X | P ∈ (2, 1)⋆ −D(X)}. Then (a) If ζ = η1 ∪ {∅} and if ∅ ≠ ζ ⊂ µ1 ∩ µ2, then (X, ζ) is a hyperconnected space. (b) If ζ = η2 ∪ {∅} and if ∅ ≠ ζ ⊂ µ1 ∩ µ2, then (X, ζ) is a hyperconnected space. D. Elgezouli et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2286-2305 2295 Proof. (a) Assume that, ζ = η1 ∪ {∅} and ∅ ̸= ζ ⊂ µ1 ∩ µ2. Let K ∈ ζ̃. Then K ∈ (1, 2)⋆ −D(X) and so c2K ∩ J ̸= ∅ for all J ∈ σ̃1. Let D ∈ ζ̃. By hypothesis, D ∈ µ1 so that D ∈ σ̃1. Thus, c2K ∩ D ̸= ∅. Since D ∈ µ2 we have K ∩ D ̸= ∅, by Lemma 3. Hence K is ζ-dense. Therefore, (X, ζ) is a hyperconnected space. (b) Suppose that, ζ = η2 ∪ {∅} and ∅ ≠ ζ ⊂ µ1 ∩ µ2. Let P ∈ ζ̃. Then P ∈ (2, 1)⋆ −D(X) and so c1P ∩ J ̸= ∅ for all J ∈ σ̃2. Let M ∈ ζ̃. By hypothesis, M ∈ µ2 so that M ∈ σ̃2. Thus, c1P ∩M ̸= ∅. Since M ∈ µ1 we have P ∩M ̸= ∅, by Lemma 3. Hence P is ζ-dense. Therefore, (X, ζ) is a hyperconnected space. Definition 22. Let (X,µ) be a GTS. A GT µ is said to satisfy the ID-property if P ∈ µ̃ and cµQ = X, then iµ(P ∩Q) ̸= ∅. Theorem 23. Let (X,µ1, µ2) be a bigeneralized topological space and η1 = {P ⊂ X | cµ1P = X}; η2 = {Q ⊂ X | cµ2Q = X}. Then (a) If ∅ ≠ ζ = η1 ∪ {∅} and if µ1 has ID-property, then (X, ζ) is a hyperconnected space. (b) If ∅ ≠ ζ = η2 ∪ {∅} and if µ2 has ID-property, then (X, ζ) is a hyperconnected space. Proof. (a) Suppose ∅ ≠ ζ = η1 ∪ {∅} and if µ1 has ID-property. Let K ∈ ζ̃. Then cµ1K = X and so K ∩ J ̸= ∅ for every J ∈ µ̃1. Take H ∈ ζ̃ which implies that H ∩M ̸= ∅ for all M ∈ µ̃1. Thus, there is D ∈ µ̃1 such that K∩D ̸= ∅ and H∩D ̸= ∅. Since cµ1H = X and D ∈ µ̃1 we have iµ1(H ∩D) ̸= ∅, by hypothesis. Thus, iµ1(H ∩D) ∈ µ̃1 which implies that K ∩ iµ1(H ∩ D) ̸= ∅ which turn implies that K ∩ H ̸= ∅. Therefore, K is ζ-dense. Hence (X, ζ) is a hyperconnected space. (b) Assume that, ∅ ≠ ζ = η2 ∪ {∅} and if µ2 has ID-property. Let L ∈ ζ̃. Then cµ2L = X and so L ∩ J ̸= ∅ for every J ∈ µ̃2. Take H ∈ ζ̃ which implies that H ∩ K ̸= ∅ for all K ∈ µ̃2. Thus, there is D ∈ µ̃2 such that L ∩D ̸= ∅ and H ∩D ̸= ∅. Since cµ2H = X and D ∈ µ̃2 we have iµ2(H ∩D) ̸= ∅, by hypothesis. Thus, iµ2(H ∩D) ∈ µ̃2 which implies that L ∩ iµ2(H ∩D) ̸= ∅ which turn implies that L ∩H ̸= ∅. Therefore, L is ζ-dense. Hence (X, ζ) is a hyperconnected space. Theorem 24. Let (X,µ1, µ2) be a bigeneralized topological space where µ1 = µ and µ2 = µ⋆⋆ ̸= ∅, µ is a generalized topology on X. Then every µ⋆⋆-dense set is (2, 1)⋆- dense set in X. Proof. Let K be a µ⋆⋆-dense set. Then c2(K) = X. By hypothesis, µ2 is a sGT. By Theorem 11, K is a (2, 1)⋆-dense set in X. Theorem 25. Let (X,µ1, µ2) satisfy the condition; if P ∈ µ̃1;Q ∈ µ̃2 and P ∩ Q ̸= ∅, then iµ1(P ∩ Q) ̸= ∅ here µ1 = µ and µ2 = µ⋆⋆ ̸= ∅ where µ is a GT on X. Then every (1, 2)⋆-dense set is µ2-dense set in X. D. Elgezouli et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2286-2305 2296 Proof. Let P ∈ (1, 2)⋆ − D(X). Then c2P ∩K ̸= ∅ for all K ∈ σ̃1. Let L ∈ µ̃2. Then L is of µ-second category and so L is not a µ-meager set which implies i1(c1(L)) ̸= ∅. Take D = i1(c1(L)). Then D ∈ µ̃1 so that D ∩ c2P ̸= ∅. Thus, c1L ∩ c2P ̸= ∅. Choose t ∈ (c1L ∩ c2P ). Then t ∈ c1L which implies H ∩ L ̸= ∅ for every H ∈ µ1(t), by Lemma 2. By hypothesis, iµ1(H ∩ L) ̸= ∅. This implies c2P ∩ iµ1(H ∩ L) ̸= ∅ which implies c2P ∩ (H ∩L) ̸= ∅ which turn implies that c2P ∩L ̸= ∅. Since L ∈ µ̃2 we have P ∩L ̸= ∅, by Lemma 3. Hence P is µ2-dense. Theorem 26. Let (X,µ1, µ2) be a BGTS here µ1 = µ and µ2 = µ⋆ where µ is a GT on X. If µ1 has the I-property, then every (1, 2)⋆-dense set is µ2-dense in X. Proof. Let Q ∈ (1, 2)⋆−D(X). Then c2Q∩K ̸= ∅ for every K ∈ σ̃1. Let L ∈ µ̃2. Then L = ⋃ t(L t 1∩Lt 2∩· · ·∩Lt nt ) where each Lt i ∈ µ̃1 for i = 1 to nt. Choose D = Lk 1∩Lk 2∩· · ·∩Lk nk for some k ; each Lk m ∈ µ̃1 for m = 1 to nk with D ̸= ∅. By hypothesis, iµ1D ̸= ∅ which implies that iµ1D ∈ µ̃1 which turn implies that iµ1D∩ c2Q ̸= ∅. Thus, c2Q∩D ̸= ∅ so that c2Q ∩ L ̸= ∅. Since L ∈ µ̃2 we have Q ∩ L ̸= ∅, By Lemma 3. Therefore, Q is µ2-dense. Theorem 27. Let (X,µ1, µ2) be a BGTS here µ1 = µ and µ2 = µ⋆⋆ where µ is a gener- alized topology on X. If (X,µ1) is a hyperconnected space and if µ1 is a sGT, then every non-null µ2-open set is (1, 2)⋆-dense in X. Proof. Let P ∈ µ̃2. Then P is of µ1-second category so that P is not a µ1-meager set which implies iµ1(cµ1(P )) ̸= ∅. Thus, iµ1(cµ1(P )) ∈ µ̃1. Let K ∈ σ̃1. By hypothesis, iµ1K ∈ µ̃1. Since (X,µ1) is a hyperconnected space we have iµ1K is µ1-dense. Therefore, iµ1(cµ1(P )) ∩ iµ1K ̸= ∅. This implies cµ1P ∩ iµ1K ̸= ∅ which implies that iµ1K ∩ P ̸= ∅, by Lemma 3. Thus, P ∩K ̸= ∅. Therefore, P ∈ (1, 2)⋆ −D(X). Theorem 28. Let (X,µ1, µ2) be a BGTS here µ1 = µ and µ2 = µ⋆ where µ is a sGT on X. If (X,µ1) is a hyperconnected space and if µ1 has I-property, then every non-null µ2-open set is a (1, 2)⋆-dense set in X. Proof. Let P ∈ µ̃⋆. Then P = ⋃ t(P t 1 ∩ P t 2 ∩ · · · ∩ P t nt ) where each P t i ∈ µ̃1 for i = 1 to nt. Choose D = P k 1 ∩P k 2 ∩· · ·∩P k nk for some k ; each P k m ∈ µ̃1 for m = 1 to nk with D ̸= ∅. By hypothesis, iµ1D ̸= ∅ which implies that iµ1D ∈ µ̃1. Since (X,µ1) is a hyperconnected space, iµ1D is µ1-dense which implies P is µ1-dense. By hypothesis and Theorem 11, P is (1, 2)⋆-dense. Theorem 29. Let (X,µ1, µ2) be a bigeneralized topological space. If (X,µs) is a hyper- connected space and if µs is a sGT for s = 1, 2, then (a) Every non-null µs-semi-open set is (s, v)⋆-dense. (b) Every non-null µs-pre-open set is (s, v)⋆-dense. (c) Every non-null µs-α-open set is (s, v)⋆-dense. (d) Every non-null µs-β-open set is (s, v)⋆-dense. D. Elgezouli et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2286-2305 2297 (e) Every non-null µs-b-open set is (s, v)⋆-dense where s, v = 1, 2 ; s ̸= v. Proof. Assume that, (X,µs) is a hyperconnected space and µs is a sGT for s = 1, 2.. Choose s = 2 and v = 1. Then (X,µ2) is a hyperconnected space, µ2 is a strong general- ized topology. (a). Let Q be a non-null µs-semi-open set. Then Q is µ2-semi-open set in X. Let H ∈ σ̃2. Suppose H ∈ µ̃2. Then there is nothing to prove. Suppose that, H /∈ µ̃2. Here H ⊂ c2(i2(H)) which implies i2(H) ∈ µ̃2, by our assumption. Since (X,µ2) is a hyperconnected space we have i2H is a µ2-dense set in X. Also, Q ⊂ c2(i2(Q)) which implies i2(Q) ∈ µ̃2 which turn implies that i2(Q)∩ i2H ̸= ∅. Thus, Q∩ i2H ̸= ∅. Therefore, c1Q ∩H ̸= ∅. Hence Q ∈ (2, 1)⋆ −D(X). (b). Let P be a non-null µs-preopen set. Then P is µ2-preopen set in X. Let G ∈ σ̃2. If G ∈ µ̃2, then the proof is trivial. Assume that, G /∈ µ̃2. Here G ⊂ c2(i2(G)) which implies i2(G) ∈ µ̃2, by our assumption which turn implies that i2G is a µ2-dense set in X. Also, P ⊂ i2(c2(P )) so that i2(c2(P )) ∈ µ̃2 for that i2(c2(P ))∩ i2G ̸= ∅. Thus, c2P ∩ i2G ̸= ∅ so that P ∩ i2G ̸= ∅, by Lemma 3. Therefore, c1P ∩G ̸= ∅. Hence P ∈ (2, 1)⋆ −D(X). (c). Let K be a non-null µs-α-open set. Then K is µ2-α-open set in X which implies K is µ2-semi-open set. Hence K ∈ (2, 1)⋆ −D(X), by (a). (d). Choose L be a non-null µs-β-open set. Then L is µ2-β-open set in X. Let M ∈ σ̃2. If M ∈ µ̃2, then there is nothing to prove. Suppose M /∈ µ̃2. Since M ⊂ c2(i2(M)) we have i2(M) ∈ µ̃2, by our assumption. By our assumption, i2M is a µ2-dense set in X. Also, L ⊂ c2(i2(c2(L))) for that i2(c2(L)) ∈ µ̃2 which turn implies that i2(c2(L)) ∩ i2M ̸= ∅. Thus, c2L ∩ i2M ̸= ∅ so that L ∩ i2M ̸= ∅, by Lemma 3. Therefore, c1L ∩M ̸= ∅. Hence L ∈ (2, 1)⋆ −D(X). (e). Take F be a non-null µs-b-open set. We get F is µ2-b-open set inX. Let V ∈ σ̃2. If V ∈ µ̃2, then the proof is obvious. Assume that, V /∈ µ̃2 then V ⊂ c2(i2(V )) so that i2(V ) ∈ µ̃2, by our assumption. Thus, i2V is a µ2-dense set in X. Here, F ⊂ c2(i2(F )) ∪ i2(c2(F )) which implies (1) i2(c2(F )) ∈ µ̃2 or (2) i2(F ) ∈ µ̃2 or (3) i2(c2(F )) ∈ µ̃2 and i2(F ) ∈ µ̃2 From the above three cases, we get F ∩ i2V ̸= ∅. Therefore, c1F ∩ V ̸= ∅. Hence F ∈ (2, 1)⋆ −D(X). By similar considerations, we can prove this theorem for the case s = 1 and v = 2. D. Elgezouli et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2286-2305 2298 Theorem 30. Let (X,µ1, µ2) be a BGTS. If (X,µs) is a hyperconnected space and if µs is a sGT for s = 1, 2, then (a) Every non-null (s, v)-µ-pre-open set is (s, v)⋆-dense. (b) Every non-null (s, v)-µ-α-open set is (s, v)⋆-dense where s, v = 1, 2 ; s ̸= v. Proof. Assume that, (X,µs) is a hyperconnected space and µs is a strong generalized topological space for s = 1, 2. Take s = 2 and v = 1. Then (X,µ2) is a hyperconnected space, µ2 is a sGT. (a). Let Q be a non-null (s, v)-µ-pre-open set where s, v = 1, 2 ; s ̸= v. Then Q is (2, 1)- µ-pre-open. Let K ∈ σ̃2. If K ∈ µ̃2, then there is nothing to prove. Assume that, K /∈ µ̃2. Here K ⊂ c2(i2(K)). By our assumption, i2K is µ2-dense. Since Q ⊂ i2(c1(Q)) we have i2(c1(Q)) ∈ µ̃2. This implies i2(c1(Q)) ∩ i2K ̸= ∅ which implies c1(Q) ∩ i2K ̸= ∅ which turn implies that c1Q ∩K ̸= ∅. Hence Q ∈ (2, 1)⋆ −D(X). (b). Take P be a non-null (s, v)-µ-α-open set where s, v = 1, 2 ; s ̸= v. We get P is (2, 1)-µ-α-open. Let G ∈ σ̃2. If G ∈ µ̃2, then the proof is trivial. Suppose G /∈ µ̃2. By our assumption, i2G is µ2-dense. Since P ⊂ i2(c1(i2(P ))) we have i2(c1(i2(P ))) ∈ µ̃2. This implies i2(c1(i2(P )))∩ i2G ̸= ∅ which implies c1(i2(P ))∩ i2G ̸= ∅ which turn implies that c1P ∩ i2G ̸= ∅. Thus, c1P ∩G ̸= ∅. Hence P ∈ (2, 1)⋆ −D(X). Similarly we can prove this theorem for the case s = 1 and v = 2. Theorem 31. Let (X,µ1, µ2) be a BGTS. If (X,µs) is a hyperconnected space, µs ⊂ µv and if µs is a strong generalized topology, then every non-null (s, v)-µ-semi-open set is (s, v)⋆-dense where s, v = 1, 2 ; s ̸= v. Proof. Assume that, (X,µs) is a hyperconnected space; µs ⊂ µv and µs is a strong generalized topological space for s = 1, 2. Take s = 1 and v = 2. Then (X,µ1) is a hyperconnected space; µ1 ⊂ µ2 and µ1 is a sGT. Let Q be a non-null (s, v)-µ-semi-open set where s, v = 1, 2 ; s ̸= v. Then Q is (1, 2)-µ- semi-open. Let H ∈ σ̃1. Suppose H ∈ µ̃1, then there is nothing to prove. Assume that, H /∈ µ̃1. Here H ⊂ c1(i1(H)). By our assumption, i1H is µ1-dense. Since Q ⊂ c2(i1(Q))) we have i1(Q) ∈ µ̃1, µ1 ⊂ µ2 and µ1 is a sGT. This implies i1(Q)∩ i1H ̸= ∅ which implies c2Q ∩H ̸= ∅. Hence Q ∈ (1, 2)⋆ −D(X). Choose s = 2 and v = 1. We get (X,µ2) is a hyperconnected space; µ2 ⊂ µ1 and µ2 is a sGT. Consider P is a non-null (s, v)-µ-semi-open set where s, v = 1, 2 ; s ̸= v. Then P is (2, 1)-µ-semi-open. Let G ∈ σ̃2. If G ∈ µ̃2, then the proof is obvious. Suppose G /∈ µ̃2. D. Elgezouli et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2286-2305 2299 Here G ⊂ c2(i2(G)). By hypothesis, i2G is µ2-dense. Since P ⊂ c1(i2(P ))) we have i2(P ) ∈ µ̃2, by hypothesis so that i2(P ) ∩ i2G ̸= ∅ which implies that c1P ∩G ̸= ∅. Hence P ∈ (2, 1)⋆ −D(X). In the rest of this section, we analyze the nature of (s, v)⋆-dense sets in a subspace. Let (X,µ) be a GTS, Q ⊂ X and µQ = {P ∩ Q | P ∈ µ}. Then µQ is called relative generalized topology on Q [7]. Theorem 32. Let (X,µ1, µ2) be a bigeneralized topological space, Q be a µs-dense sub- space of X for s = 1, 2. If P is a µsQ-dense and µs is a sGT, then P ∈ (s, v)⋆ − D(X) where s, v = 1, 2 ; s ̸= v. Proof. Assume that, Q is µs-dense in X and µs is a strong generalized topology for s = 1, 2. Let P be a µsQ-dense set in Q where s = 1, 2. Take s = 1 and v = 2. Then Q is µ1-dense, µ1 is a sGT and P is µ1Q-dense in Q. Let K ∈ σ̃1. If K ∈ µ̃1, then further proof investigation no longer required. Suppose K /∈ µ̃1. By hypothesis, iµ1K ∈ µ̃1 which implies iµ1K ∩ Q ∈ ˜µ1Q . Take L = iµ1K ∩ Q. Then L ∩ P ̸= ∅ so that iµ1K ∩ P ̸= ∅. This implies K ∩ P ̸= ∅ which implies K ∩ c2P ̸= ∅. Therefore, P ∈ (1, 2)⋆ −D(X). Fix s = 2, v = 1. Then Q is µ2-dense, µ2 is a sGT and P is µ2Q-dense in Q. Let M ∈ σ̃2. If M ∈ µ̃2, then the proof is directly follows. Assume that, M /∈ µ̃2. By hypothesis, iµ2M ∈ µ̃2 which implies iµ2M ∩Q ∈ ˜µ2Q . Take V = iµ2M ∩Q. Then V ∩ P ̸= ∅ so that iµ2M ∩ P ̸= ∅ which implies M ∩ P ̸= ∅ which turn implies that M ∩ c2P ̸= ∅. Hence, P ∈ (2, 1)⋆ −D(X). Theorem 33. Let (X,µ1, µ2) be a BGTS, µs satisfy the I-property and Q be a µs-open subset of X for s = 1, 2. If µs ⊂ µv and if P ∈ (s, v)⋆ −D(X), then P is µsQ-dense set in Q where P ⊂ Q; s, v = 1, 2 ; s ̸= v. Proof. Assume that, Q is µs-open subset of X;µs ⊂ µv and P ∈ (s, v)⋆ − D(X) for s, v = 1, 2 ; s ̸= v. Choose s = 1 and v = 2. Then Q ∈ µ̃1 ; µ1 ⊂ µ2 and P ∈ (1, 2)⋆ − D(X). Let L ∈ ˜µ1Q . Then L = K ∩Q where K ∈ µ̃1. By hypothesis, iµ1L ∈ µ̃1. This implies L∩ c2P ̸= ∅ which implies that L ∩ P ̸= ∅, by Lemma 3. Hence P is a µ1Q-dense set in Q. Fix s = 2 and v = 1. We get Q ∈ µ̃2 ; µ2 ⊂ µ1 and P ∈ (2, 1)⋆ − D(X). Let V ∈ ˜µ2Q . Then V = M ∩ Q where M ∈ µ̃2. By assumption, iµ2V ∈ µ̃2 so that V ∩ c1P ̸= ∅ which implies that V ∩ P ̸= ∅, by Lemma 3. Therefore, P is a µ2Q-dense set in Q. D. Elgezouli et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2286-2305 2300 Theorem 34. Let (X,µ1, µ2) be a bigeneralized topological space, Q be a µs-open subset of X and µs satisfy the I-property for s = 1, 2. If µsQ is a sGT and P ∈ (s, v)⋆ −D(X), then P ∈ (µsQ , µv)−D(Q) where P ⊂ Q; s, v = 1, 2 ; s ̸= v. Proof. Suppose that, Q ∈ µ̃s, µs satisfy the I-property and µsQ is a strong generalized topology for s = 1, 2. Let P ∈ (s, v)⋆ −D(X) where s, v = 1, 2 ; s ̸= v. Choose s = 1 and v = 2. Then Q ∈ µ̃1, µ1 satisfy the I-property, µ1Q is a sGT and P ∈ (1, 2)⋆ − D(X). Let J ∈ σ̃1Q . If J ∈ µ̃1Q , then there is nothing to prove. Sup- pose J /∈ µ̃1Q . Since J ∈ σ̃1Q and µ1Q is a strong subspace generalized topology we have i1QJ ∈ µ̃1Q . Take K = i1QJ. Then K ̸= ∅ and K = L ∩Q where L ∈ µ̃1. Since L,Q ∈ µ̃1 and µ1 satisfy the I-property, iµ1(K) ∈ µ̃1. This implies iµ1K ∩ c2P ̸= ∅ which implies K ∩ c2P ̸= ∅ which turn implies that J ∩ c2P ̸= ∅. Hence P ∈ (µ1Q , µ2) ⋆ −D(Q). Take s = 2 and v = 1. We get Q ∈ µ̃2, µ2 satisfy the I-property, µ2Q is a sGT and P ∈ (2, 1)⋆ −D(X). Let V ∈ σ̃2Q . If V ∈ µ̃2Q , then the proof is obvious. Assume V /∈ µ̃2Q . by the definition of V and µ2Q is a strong subspace generalized topology we have i2QV ∈ µ̃2Q . Take L = i2QV. Then L ̸= ∅ and L = M ∩ Q where M ∈ µ̃2. Here, M,Q ∈ µ̃2 and µ2 satisfy the I-property, iµ2(L) ∈ µ̃2 so that iµ2L∩c1P ̸= ∅ which implies L∩c1P ̸= ∅ which turn implies that V ∩ c1P ̸= ∅. Therefore, P ∈ (µ2Q , µ1) ⋆ −D(Q). Theorem 35. Let (X,µ1, µ2) be a BGTS and Q be a µs-dense subset of X for s = 1, 2. If µs is a strong generalized topology and if P ∈ (µsQ , µvQ) ⋆−D(Q), then P ∈ (s, v)⋆−D(X) for s, v = 1, 2 ; s ̸= v. Proof. Assume that, P ∈ (µsQ , µvQ) ⋆ −D(Q) where s, v = 1, 2 ; s ̸= v. Choose s = 1 and v = 2. Then P ∈ (µ1Q , µ2Q) ⋆ − D(Q). Let H ∈ σ̃1. Suppose H ∈ µ̃1. Then H ∩ Q ∈ µ̃1Q . Take K = H ∩ Q. Then K ∩ c2QP ̸= ∅ so that K ∩ c2P ̸= ∅. This implies H ∩ c2(P ) ̸= ∅ which implies that P ∈ (1, 2)⋆ −D(X). If H /∈ µ̃1, then i1H ∈ µ̃1. Take L = i1H. Then by similar arguments in the above case, we get P ∈ (1, 2)⋆ −D(X). Fix s = 2 and v = 1. We get P ∈ (µ2Q , µ1Q) ⋆ −D(Q). Let G ∈ σ̃2. Suppose G ∈ µ̃2 we get G ∩Q ∈ µ̃2Q . Choose K = G ∩Q so that K ∩ c1QP ̸= ∅ which implies that K ∩ c1P ̸= ∅. Thus, G ∩ c1(P ) ̸= ∅ so that P ∈ (2, 1)⋆ − D(X). Assume that, G /∈ µ̃2, then i2G ∈ µ̃2. Take L = i2G. By similar considerations, we get P ∈ (2, 1)⋆ −D(X). 4. Images of (s, v)⋆-dense sets A function f : (X,µ) → (Y, η) is said to be (µ, η)-continuous [4] (resp. (µ, η)-open) [18] if f−1(Q) ∈ µ whenever Q ∈ η (resp. f(P ) ∈ η whenever P ∈ µ). D. Elgezouli et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2286-2305 2301 Lemma 5. [12, Lemma 7.3] A map f : (X,µ) → (Y, η) is (µ, η)-open if and only if f−1(cP ) ⊂ c(f−1(P )) for any P ⊂ Y. Theorem 36. Let (X,µ1, µ2) and (Y, η1, η2) be two BGTSs. If f : X → Y is (µt, ηt)- continuous for t = 1, 2 and ηs is sGT for s = 1, 2, then image of a (s, v)⋆-dense set is (s, v)⋆-dense where s, v = 1, 2 ; s ̸= v. Proof. Assume that, f is (µt, ηt)-continuous for t = 1, 2. Let Q ∈ (s, v)⋆−D(X) where s, v = 1, 2 ; s ̸= v. Fix s = 1 and v = 2. We get Q ∈ (1, 2)⋆ − D(X) so that cµ2Q ∩H ̸= ∅ for H ∈ σ̃µ1 . Let K ∈ σ̃η1 . By assumption, η1 is a sGT so that iη1K ∈ η̃1. This implies f−1(iη1K) ∈ µ̃1, by hypothesis which implies that cµ2Q ∩ f−1(iη1K) ̸= ∅. Thus, f(cµ2Q ∩ f−1(iη1K)) ̸= ∅ so that f(cµ2Q) ∩ iη1K ̸= ∅. Since f is (µ1, η1)-continuous we have cη2(f(Q)) ∩ iη1K ̸= ∅. Therefore, f(Q) ∈ (1, 2)⋆ −D(Y ). Take s = 2 and v = 1. Then Q ∈ (2, 1)⋆−D(X) and so cµ1Q∩M ̸= ∅ for M ∈ σ̃µ2 . Choose L ∈ σ̃η2 . By hypothesis, η2 is a sGT so that iη2L ∈ η̃2 which implies f−1(iη2L) ∈ µ̃2, by assumption which turn implies that cµ1Q∩ f−1(iη2L) ̸= ∅. Thus, f(cµ1Q∩ f−1(iη2L)) ̸= ∅ for that f(cµ1Q) ∩ iη2L ̸= ∅. By hypothesis, cη1(f(Q)) ∩ iη2L ̸= ∅. Hence f(Q) ∈ (2, 1)⋆ − D(Y ). Theorem 37. Let (X,µ1, µ2) and (Y, η1, η2) be two bigeneralized topological spaces. If f : X → Y is (µt, ηt)-open for t = 1, 2 ; one-one map and µs is sGT for s = 1, 2, then inverse image of a (s, v)⋆-dense set is (s, v)⋆-dense. Proof. Let P ∈ (s, v)⋆ −D(Y ) for s, v = 1, 2 ; s ̸= v. Fix s = 1 and v = 2. Then P ∈ (1, 2)⋆ − D(Y ) so that cη2P ∩ L ̸= ∅ for all L ∈ σ̃η1 . Let D ∈ σ̃µ1 so that iµ1D ∈ µ̃1, by assumption. Since f is (µ1, η1)-open we have f(iµ1D) ∈ η̃1. This implies cη2P ∩f(iµ1D) ̸= ∅ which implies that f−1(cη2P )∩f−1(f(iµ1D)) ̸= ∅. Here f is an injective map, f−1(cη2P ) ∩ iµ1D ̸= ∅. By Lemma 5, cµ2(f −1(P )) ∩ ıµ1D ̸= ∅. Hence f−1(P ) ∈ (1, 2)⋆D(X). Choose s = 2 and v = 1. We get P ∈ (2, 1)⋆ − D(Y ) implies that cη1P ∩ M ̸= ∅ for all M ∈ σ̃η2 . Choose V ∈ σ̃µ2 so that iµ2V ∈ µ̃2, by hypothesis which implies f(iµ2V ) ∈ η̃2. Thus, cη1P ∩ f(iµ2V ) ̸= ∅ so that f−1(cη1P ) ∩ f−1(f(iµ2V )) ̸= ∅. Since f is an injective map, f−1(cη1P ) ∩ iµ2V ̸= ∅. By Lemma 5, cµ1(f −1(P )) ∩ ıµ2V ̸= ∅. Therefore, f−1(P ) ∈ (2, 1)⋆ −D(X). 5. Applications for (s, v)⋆-dense sets In 1999, Molodstov introduced a new mathematical tool namely, soft set theory [14]. It has been used for dealing with uncertainty. Most of the researchers presented an appli- D. Elgezouli et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2286-2305 2302 cation 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 some subsets defined in a bigeneralized topological space and also in generalized topological space. Example 38. Consider the BGTS (X,µ1, µ2) where X = {a, b, c, d}; µ1 = {∅, {b}, {a, d}, {b, d}, {a, b, d}}; and µ2 = {∅, {c}, {a, b}, {a, c}, {a, b, c}}. Here, • σ1 = {∅, {b}, {c}, {a, d}, {b, c}, {b, d}, {a, b, d}, {a, c, d}, {b, c, d}, X}. • σ2 = {∅, {c}, {d}, {a, b}, {a, c}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, X}. Then we get, • (1, 2)⋆ −D(X) = {{a, c}, {b, c}, {a, b, c}, {a, c, d}, {b, c, d}, X}. • (2, 1)⋆−D(X) = {{d}, {a, b}, {a, d}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}, X}. Let U = {a, c, d} be a subset of X and E = {(1, 2)⋆-dense set, (2, 1)⋆-dense set, (1, 2)⋆- dense but not (2, 1)⋆-dense, (2, 1)⋆-dense but not (1, 2)⋆-dense, (1, 2)⋆-dense and (2, 1)⋆- dense } = {e1, e2, e3, e4, e5} is the set of parameters. Define a map F from E to exp(U) by, F (e1) = {a, c};F (e2) = {d};F (e3) = {a, c};F (e4) = {c, d}, F (e5) = {a, c, d}. Then the pair (F,E) is a soft set over U. Example 39. Consider the BGTS (X,µ1, µ2) where X = {p, q, r, s}; µ1 = {∅, {p}, {p, s}, {q, s}, {p, q, s}} and µ2 = {∅, {q}, {p, r}, {q, r}, {p, q, r}}. Here, • µ1-semi-open sets = {∅, {p}, {r}, {p, r}, {p, s}, {q, s}, {p, q, s}, {p, r, s}, {q, r, s}, X}. • µ1-pre-open sets = {∅, {p}, {s}, {p, q}, {p, s}, {q, s}, {p, q, s}}. • µ1 − α-open sets = {∅, {p}, {p, s}, {q, s}, {p, q, s}}. • µ1 − β-open sets = exp(X)− {{q}, {q, r}}. • µ1 − b-open sets = exp(X)− {q}. Let U = {q, r, s} be a subset ofX and E = {µ1-semi-open set, µ1-pre-open set, µ1−α-open set, µ1−β-open set, µ1− b-open set } = {e1, e2, e3, e4, e5} is the set of parameters. Define a function F from a set E to exp(U) by, F (e1) = {r};F (e2) = {s};F (e3) = {q, s};F (e4) = D. Elgezouli et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2286-2305 2303 {r, s};F (e5) = {q, r}. Then the pair (F,E) is a soft set over U. Here, • µ2-semi-open sets = {∅, {q}, {s}, {p, r}, {q, r}, {q, s}, {p, q, r}, {p, r, s}, {q, r, s}, X}. • µ2-pre-open sets = {∅, {q}, {r}, {p, q}, {p, r}, {q, r}, {p, q, r}}. • µ2 − α-open sets = {∅, {q}, {p, r}, {q, r}, {p, q, r}}. • µ2 − β-open sets = exp(X)− {{p}, {p, s}}. • µ2 − b-open sets = exp(X)− {{p}, {p, s}}. Let U = {p, r, s} be a subset ofX and E = {µ2-semi-open set, µ2-pre-open set, µ2−α-open set, µ2−β-open set, µ2− b-open set } = {e1, e2, e3, e4, e5} is the set of parameters. Define a function F from a set E to exp(U) by, F (e1) = {s};F (e2) = {r};F (e3) = {q};F (e4) = {r, s};F (e5) = {p, r}. Then the pair (F,E) is a soft set over U. Example 40. Consider the BGTS (X,µ1, µ2) where X = {p, q, r, s}; µ1 = {∅, {r}, {p, s}, {r, s}, {p, r, s}} and µ2 = {∅, {q}, {q, s}, {r, s}, {q, r, s}}. Here, • (s, v)-µ1-regular open sets = {∅, {r}, {p, r, s}}. • (s, v)-µ1-semi-open sets = {∅, {p}, {r}, {p, r}, {p, s}, {r, s}, {p, r, s}}. • (s, v)-µ1-pre-open sets = {∅, {r}, {s}, {p, s}, {r, s}, {p, r, s}, }. • (s, v)-µ1-α-open sets = {∅, {r}, {p, s}, {r, s}, {p, r, s}}. Let U = {p, r, s} be a subset of X and E = {(s, v)-µ1-regular open, (s, v)-µ1-semi-open set, (s, v)-µ1-pre-open set, (s, v)-µ1 − α-open set } = {e1, e2, e3, e4, } is the set of parameters. Define a map F from a non-null set E to exp(U) by, F (e1) = {r};F (e2) = {p};F (e3) = {s};F (e4) = {r, s}. Then the pair (F,E) is a soft set over U. Now, • (s, v)-µ2-regular open sets = {{q}, {q, s}, {q, r, s}}. • (s, v)-µ2-semi-open sets = {∅, {q}, {q, s}, {r, s}, {p, q, s}, {p, r, s}, {q, r, s}, X}. • (s, v)-µ2-pre-open sets = {∅, {q}, {s}, {q, s}, {r, s}, {q, r, s}, }. • (s, v)-µ2-α-open sets = {∅, {q}, {q, s}, {r, s}, {q, r, s}}. Let U = {q, r, s} be a subset of X and E = {(s, v)-µ2-regular open, (s, v)-µ2-semi-open set, (s, v)-µ2-pre-open set, (s, v)-µ2 − α-open set } = {e1, e2, e3, e4, } is the set of param- eters. Define a map F from a set E to exp(U) by, F (e1) = {q};F (e2) = {q, s};F (e3) = {s};F (e4) = {r, s}. Then the pair (F,E) is a soft set over U. REFERENCES 2304 6. Conclusion In this article, we are given additional tricks for finding the significance of a given set in a bigeneralized topological space. Also, we have proven some results for checking whether the given set is (s, v)⋆-dense or not. Finally, we defined soft sets using various open sets and (s, v)⋆-dense sets. References [1] D. Andrijević. 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