EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 4, 2022, 1869-1886 ISSN 1307-5543 – ejpam.com Published by New York Business Global Special Sequences On Generalized Metric Spaces Yasser Farhat1, Amel Souhail2, Vadakasi Subramanian3,∗, K. Muthumari3 1 Academic Support Department, Abu Dhabi Polytechnic, P. O. Box 111499, Abu Dhabi, UAE 2 Départment de Mathématiques, Faculté des Sciences de Sfax, Université de Sfax, Route de Soukra Km 3.5, B.P. 1171, 3000, Sfax, Tunisia 3 Department of Mathematics, A.K.D. Dharma Raja Women’s College, Rajapalayam, Tamilnadu, India Abstract. In this article, in a generalized metric space, we will focus on new types of sequences. We introduce three new kinds of Cauchy sequences and study their significance in generalized metric spaces. Also, we give several interesting properties of these sequences. 2020 Mathematics Subject Classifications: 40A05, 54E35, 54E40, 54E50 Key Words and Phrases: Kernel, Bourbaki-Cauchy sequence, pseudo-Cauchy sequence, Bourbaki- complete metric space, weakly complete metric space and asymptotic sequence 1. Introduction The notion of completeness plays in the theory of metric spaces governs the im- portance upgrade in topological metrics. Few researchers had examined the significance of complete metric spaces e.g. [4, 5, 8, 10–13, 15–19]. Inspiring the earlier literature, Korczak-Kubiak et.al introduced the concept of new metric space namely, generalized metric space during 2013 and then defined two new things in the name of kernel and per- fect kernel in a generalized metric space. Using these tools, they generated three types of complete spaces which has been defined weakly complete, complete and strongly complete spaces. In addition,they have analyzed the nature of these three types of complete spaces in generalized metric spaces. Similarly, the nature of Bourbaki-Cauchy sequences e.g. [2, 3, 9] has been obtained which influenced some new types of sequences namely Bourbaki-Cauchy, co-finally-Cauchy and pseudo-Cauchy in a metric space given by Aggarwal et.al [1]. Motivated by this, in a generalized metric space, we redefined three types of sequences namely, Bourbaki-Cauchy, cofinally-Cauchy and pseudo-Cauchy sequences as well as com- plete metric spaces. Also, we investigate their significance and give some relationships between these sequences in a generalized metric space. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4592 Email addresses: farhat.yasser.1@gmail.com (Y. Farhat),Vadakasisub1992@gmail.com (S. Vadakasi) https://www.ejpam.com 1869 © 2022 EJPAM All rights reserved. V. Subramanian et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1869-1886 1870 Throughout this paper, R and N denote the set of all real and natural numbers, respectively. 2. Preliminaries First, we recall some basic things defined in a generalized metric space which are useful to the development of the next sections. Let X ̸= ∅. The symbol Ω to denote the family consisting of metrics defined on subsets of X, that is σ ∈ Ω, then there exists a non-null set Bσ ⊂ X such that σ is a metric on Bσ where Bσ is a domain space of σ and it will be denoted by dom(σ). Then (X,Ω) is called as a generalized metric space (GMS) [7]. If σ is a metric on X and C is a non-null subset X, then we write σ|C for the restriction of the metric σ to C × C. Moreover, put Ω|C = {σ|C | σ ∈ Ω} for any Ω ⊂ ΩX where ΩX is the collection of all metrics defined on X [7]. Obviously, if (X,ΩX) is a generalized metric space, Ω ⊂ ΩX and C is a non-null subset of X, then (C,Ω|C) is a generalized metric space [7]. Denote µΩ is the family of Ω-open sets in a generalized metric space (X,Ω), more precisely, H ∈ µΩ if and only if for each c ∈ H, there exist σ ∈ Ω and ε > 0 such that Bσ(c, ε) ⊂ H where Bσ(c, ε) = {d ∈ dom(σ) | σ(c, d) < ε} [7]. Let X be any non-null set. A collection µ of subsets of X is a generalized topology [7] in X if it contains the empty set and it closed under arbitrary union. Then the pair (X,µ) is called as a generalized topological space (GTS) [7].The elements of µ are called µ-open set of X. Moreover, if (X,Ω) is a GMS, then (X,µΩ) is a GTS [7]. Now we remember some definitions and lemmas that are found in [7]. Definition 1. A subset Q of X is said to be µ-dense if cµ(Q) = X. Definition 2. Let (X,Ω) be a GMS. A finite family Ω0 ⊂ Ω is called a kernel of the space X if for any J ∈ µ̃Ω, there exists σ ∈ Ω0 such that iσ(J) ̸= ∅ where µ̃Ω = {K ∈ µ̃Ω|K ̸= ∅}. Definition 3. Let (X,Ω) be a GMS and {zn}n∈N be a sequence in X. Then {zn} is said to be : (i) Cauchy if there is a metric σ ∈ Ω and for every ε > 0, there exists a positive integer N0 such that σ(zn, zm) < ε for all n,m ≥ N0. (ii) convergent to z ∈ X if there is a metric σ ∈ Ω and for every ε > 0, there exists a positive integer N1 such that σ(zn, z) < ε for all n,m ≥ N1. Then z is called the limit point of the sequence {zn}n∈N. Definition 4. A generalized metric space (X,Ω) is a weakly complete metric space if there exists a kernel Ω0 ⊂ Ω consisting of complete metrics. Definition 5. A space X is called as a hyperconnected space [6] if every non-null µ-open subset H of X is a µ-dense set. V. Subramanian et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1869-1886 1871 3. Relationship among sequences Here, we degenerate three kinds of sequences namely, Bourbaki-Cauchy, pseudo- Cauchy and cofinally-Cauchy sequences in generalized metric spaces. In a generalized metric space, the nature of these sequences is analyzed. In view of complete metric spaces founded in [7], we define three types of complete metric spaces, namely Bourbaki-complete, pseudo-complete and cofinally-complete metric spaces. The relationship between these complete spaces is explored. Definition 6. Let (X,Ω) be a generalized metric space and ε be a positive number. An ordered set of points {c0, c1, ..., cm} is said to be ε-chain of length of m from c0 to cm if there is a metric σ ∈ Ω satisfying σ(ci−1, ci) < ε where i = 1 to m. Example 7. Consider the generalized metric space (X,Ω) where X = R,Ω = {σ1, σ2, σ3} and the metrics are defined by σ1(c, d) = |c− d|; σ2(c, d) = { 0 if c = d, 1 if c ̸= d. and σ3(c, d) = σ1(c,d) 1+σ1(c,d) for all c, d ∈ X. (a). Let ε = 0.5 and {cm}m=13 m=0 be a set of points where c0 = 1; cm = cm−1 + 0.01 for m = 1 to 13. Then the set of points {cm}m=13 m=0 have a ε-chain of length of m from c0 to cm with respect to σ1, σ2 and σ3. (b). Let ε = 0.6 and {cn}n∈N = { 1 n}n∈N be a sequence in X. Take {cm}m=13 m=0 be a set of points where c0 = 1; cm = c0 m+1 for m = 1 to 13. Then the set of points {cm}m=13 m=0 have a ε-chain of length of m from c0 to cm with respect to σ1. Definition 8. Let (X,Ω) be a generalized metric space. A sequence {zn}n∈N is said to be: (a) Bourbaki-Cauchy with respect to σ ∈ Ω in X if for every ε > 0, there exist r ∈ N and n0 ∈ N such that whenever n > j ≥ n0 the points zj and zn can be joined by an ε-chain of length r with respect to the same metric. (b) pseudo-Cauchy with respect to σ ∈ Ω in X if for every ε > 0 and for every n ∈ N, there exist s, t ∈ N, s ̸= t such that s, t > n and σ(zs, zt) < ε. (c) cofinally-Cauchy with respect to σ ∈ Ω in X if for every ε > 0, there exists an infinite subset Nε of N such that for every l,m ∈ Nε we have σ(zl, zm) < ε. We notated by, B(σ) = {{dn} | {dn} is a Bourbaki-Cauchy sequence with respect to σ in X}; P(σ) = {{ln} | {ln} is a pseudo-Cauchy sequence with respect to σ in X}; C(σ) = {{ln} | {ln} is a cofinally-Cauchy sequence with respect to σ in X}; C(σ) = {{un} | {un} is a Cauchy sequence with respect to σ in X} where σ ∈ Ω. V. Subramanian et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1869-1886 1872 Definition 9. Let (X,Ω) be a GMS. Then σ ∈ Ω is called Bourbaki (resp. pseudo, cofi- nally) complete metric if every Bourbaki-Cauchy (resp. pseudo-Cauchy, cofinally-Cauchy) sequence with respect to σ is a convergent sequence with respect to σ. Definition 10. A GMS (X,Ω) is said to be a Bourbaki (resp. pseudo, cofinally) complete metric space if there exists a kernel Ω0 ⊂ Ω consisting of all Bourbaki (resp. pseudo, cofinally) complete metrics. Theorem 11 and Example 12 are describe the below diagram. Cauchy sequence Bourbaki− Cauchy sequence / Theorem 11 provides an easy way to check, in a generalized metric space, whether a given sequence is Bourbaki-Cauchy or not. This theorem is direct impact of the definitions (Definition 3 and Definition 8) so the trivial proof is neglected. Theorem 11. Let (X,Ω) be a generalized metric space. Then the followings are true. (a) Every Cauchy sequence is a Bourbaki-Cauchy sequence with the same metric. (b) Every Bourbaki -complete metric space is a weakly complete metric space. Example 12. Consider the generalized metric space (X,Ω) where X = R,Ω = {σm | m ∈ N} ⊂ ΩX with σm(c, d) = min{σE(c, d), 1 m} where σE(c, d) = |c− d|. Let {cn}n∈N = {n+ 0.1 8 } and ε > 0 be given. Then {cn} ∈ B(σ1). But {cn} /∈ C(σ1). Cauchy sequence cofinally − Cauchy sequence / The following Theorem 13 and Example 14 are describe the above diagram. The following Theorem 13 gives a shortcut for finding the nature of a given Cauchy sequence in a generalized metric space. Theorem 13. Let (X,Ω) be a generalized metric space. Then the followings are true. (a) Every Cauchy sequence is a cofinally-Cauchy sequence with the same metric. (b) Every cofinally-complete metric space is a weakly complete metric space. Proof. (a). Let σ ∈ Ω and {cn}n∈N ∈ C(σ). Let ε > 0 be given. Then there exists a positive integer N0 such that σ(cn, cm) < ε for all n,m ≥ N0. Take Nε = {j : j ≥ N0}. Then Nε is an infinite subset of N and σ(ck, cl) < ε for all k, l ∈ Nε. Hence {cn} ∈ C(σ). (b). Directly follows from (a) and the definition of weakly complete metric space. V. Subramanian et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1869-1886 1873 Example 14. Consider the generalized metric space (X,Ω1) where X = R+ ∪ {0},Ω1 = {σ1, σ2, σ3} ⊂ ΩX and the metrics are defined by σ1(c, d) = |c− d|; σ2(c, d) = { 0 if c = d, c+ d if c ̸= d. and σ3(c, d) = σ1(c,d) 1+σ1(c,d) for all c, d ∈ X. Define a sequence {cn}n∈N by {cn} = { log(n) if n is prime, 0 otherwise. in X. Then {cn} ∈ C(σ2). But for every ε > 0, there is no N0 ∈ N such that σ2(cn, cm) < ε for n,m ≥ N0. Thus, {cn} /∈ C(σ2). Cauchy sequence pseudo− Cauchy sequence / The following Theorem 15 and Example 16 are describe the above diagram. Theorem 15 is reduce the complexity for finding, in a generalized metric space, whether a given Cauchy sequnce is pseudo-Cauchy sequence or not. Theorem 15. Let (X,Ω) be a generalized metric space. Then the followings are true. (a) Every Cauchy sequence is a pseudo-Cauchy sequence in X with the same metric. (b) Every pseudo-complete metric space is a weakly complete metric space. Proof. (a). Let {cn}n∈N ∈ C(σ). Then there is N0 ∈ N such that σ(cn, cm) < ε for all n,m ≥ N0. Let ε > 0, k ∈ N be given. If k ≥ N0, then the result is obvious. Suppose that k < N0. Since N0 ∈ N, we get elements l, j ∈ N, l ̸= j such that l, j > N0 > k. Also, σ(cl, cj) < ε. Therefore, {cn} ∈ P(σ). (b). Follows from (a). Example 16. Consider the generalized metric space (X,Ω1) whereX = R,Ω1 = {σ1, σ2, σ3} ⊂ ΩX and the metrics are defined by σ1(c, d) = { 0 if c = d, 1 if c ̸= d. σ2(c, d) = |c− d| and σ3(c, d) = min{1, σ1(c, d)} for all c, d ∈ X. Define a sequence {cn}n∈N by {cn} = {1, 1, 2, 2, 3, 3, 4, 4, 5, 5, .....} in X. Then {cn} ∈ P(σ2). But for every ε > 0, there exists no N0 ∈ N such that σ2(cn, cm) < ε for n,m ≥ N0. Thus, {cn} /∈ C(σ2). V. Subramanian et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1869-1886 1874 Theorem 17 and Example 18 are describe the below diagram. The below Theorem 17 gives the relations between Bourbaki-Cauchy sequence and cofinally-Cauchy sequence in a GMS. Bourbaki− Cauchy sequence cofinally − Cauchy sequence / Theorem 17. Let (X,Ω) be a GMS. Then the followings are true. (a) Every Bourbaki-Cauchy sequence is a cofinally-Cauchy sequence in X with the same metric. (b) Every cofinally-complete metric space is a Bourbaki -complete metric space. Proof. Let {cn}n∈N ∈ B(σ) and ε > 0 be given. Then there is m,n0 ∈ N such that whenever n > j ≥ n0, the points cj and cn can be joined by an ε-chain of length m. Let Nε = {n | n ≥ n0 and cn, cj can be joined by ε m -chain of length m whenever m − 1 points lie between cn and cj where n ̸= j; j ≥ n0;m, j ∈ N}. Then Nε is an infinite subset of N. Let k, l ∈ Nε. If k and l are consecutive numbers, then there is nothing to prove. Suppose that there is some p points lies between k and l. Then ck and cl can be joined by ε p+1 -chain of length p+ 1 and so the distance between the consecutive numbers is less than ε p+1 . Thus, σ(ck, cl) < ε. Hence {cn} ∈ C(σ). (b). Directly follows from the definition of Bourbaki-complete metric space and by (a). Example 18. Consider the generalized metric space (X,Ω1) whereX = R,Ω1 = {σ1, σ2, σ3} ⊂ ΩX and then the metrics are defined by σ1(c, d) = min{|c− d|, 1}; σ2(c, d) = { 0 if c = d, 1 if c ̸= d and hence σ3(c, d) = σ1(c, d)+σ2(c, d) for all c, d ∈ X. Define a sequence {cn}n∈N by {cn} = {1,−1, 2,−1, 3,−1, 4,−1, .....} in X. Then {cn} ∈ C(σ1). But {cn} /∈ B(σ1). Because, for every ε > 0, there is no m,n0 ∈ N such that whenever n > j ≥ n0, the points cj and cn can be joined by an ε-chain of length m. Theorem 19. Let (X,Ω) be a GMS. Then every cofinally-Cauchy sequence has a Cauchy subsequence with the same metric. Proof. Let σ ∈ Ω and {cn}n∈N ∈ C(σ). Given ε > 0. Then there is an infinite subset Nε of N such that σ(cl, ck) < ε for each l, k ∈ Nε. Let A be a partially ordered subset of Nε with < ordering and {cnk } be a subsequence of {cn} where nk ∈ A. Choose M = min{ni | ni ∈ A}. For np, nq > M, we get σ(cnp , cnq) < ε, by the definition of A. Thus, σ(cnp , cnq) < ε for every np, nq > M. Thus, {cnk } ∈ C(σ). Therefore, {cn} has a subsequence in C(σ). Theorem 20 and Example 21 are describe the below diagram. V. Subramanian et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1869-1886 1875 cofinally − Cauchy sequence pseudo− Cauchy sequence / Theorem 20 provides easier way to explore, in a generalized metric space, a given sequence is pseudo-Cauchy or not. Theorem 20. Let (X,Ω) be a GMS. Then every cofinally-Cauchy sequence is a pseudo- Cauchy sequence with the same metric. Proof. Let {cn} ∈ C(σ) and ε > 0 be given. Then there exists an infinite subset Nε of N such that σ(ck, cl) < ε for every k, l ∈ Nε. Let n ∈ N. Since Nε is an infinite subset of N, there exist p, q ∈ Nε with p ̸= q and p, q > n. Also, σ(cp, cq) < ε. Hence {cn} ∈ P(σ). Example 21 shows that the reverse part of Theorem 20 is need not be true. Example 21. Consider the generalized metric space (X,Ω1) whereX = R,Ω1 = {σ1, σ2, σ3} ⊂ ΩX such that σ1(c, d) = { 0 if c = d, 1 if c ̸= d σ2(c, d) = σE(c, d) where σE(c, d) = |c − d| and then σ3(c, d) = σ1(c,d) 1+σ1(c,d) for all c, d ∈ X. Define a sequence {cn}n∈N = {1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, ....}. Then {cn} ∈ P(σ2). But there is no infinite subset Nε of N such that σ(ck, cl) < ε for every ε > 0 and k, l ∈ Nε. Thus, {cn} /∈ C(σ2). Next, in the rest of this section, in a generalized metric space, we define a rearrangement of a sequence and give some interesting results based on this definition. Definition 22. Let (X,Ω) be a generalized metric space. Then the sequence {dn} is a rearrangement of a sequence {cn} if there is a 1-1 correspondence h : N → N such that n ∈ N, dn = ch(n). Theorem 23. Let (X,Ω) be a GMS. If {cn} ∈ C(σ), then every rearrangement sequence of {cn} is cofinally-Cauchy with the same metric. Proof. Suppose {cn} ∈ C(σ). Let ε > 0 be given. Then there exists an infinite subset Nε of N such that σ(ck, cl) < ε for every k, l ∈ Nε. Let {dn} be a rearrange- ment of a sequence {cn}. Then there is a 1-1 correspondence h : N → N such that n ∈ N, dn = ch(n). Take Nε0 = {h−1(n)|n ∈ Nε}. Then Nε0 is an infinite subset of N. Let p, q ∈ Nε0 . Then p = h−1(s) and q = h−1(t) where s, t ∈ Nε. Now σ(dp, dq) = σ(dh−1(s), dh−1(t)) = σ(ch(h−1(s)), ch(h−1(t))) which implies σ(dp, dq) = σ(cs, ct), since h is bijective. Thus, σ(dp, dq) < ε, since s, t ∈ Nε. Therefore, {dn} ∈ C(σ). Since {dn} is an arbitrary rearrangement sequence of {cn} we have every rearrangement sequence of {cn} is cofinally-Cauchy with the same metric. V. Subramanian et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1869-1886 1876 Theorem 24. Let (X,Ω) be a GMS. If the rearrangement sequence of {cn} is cofinally- Cauchy, then {cn} is cofinally-Cauchy with the same metric. Proof. Assume that, the rearrangement sequence {dn} of {cn} is in C(σ). Then there exists an infinite subset Nε of N such that σ(dl, dm) < ε for every l,m ∈ Nε. By as- sumption, there is a 1-1 correspondence h : N → N such that n ∈ N, dn = ch(n). Take Nε1 = {h(n)|n ∈ Nε}. Then Nε1 is an infinite subset of N. Let n,m ∈ Nε1 . Then n = h(i) and m = h(j) where i, j ∈ Nε. Now σ(cn, cm) = σ(ch(i), ch(j)) which implies σ(cn, cm) = σ(di, dj). Thus, σ(cn, cm) < ε, since i, j ∈ Nε. Therefore, {cn} ∈ C(σ). Theorem 25. Let (X,Ω) be a GMS, h : N → N be a strictly increasing, bijective function. If the rearrangement sequence of {cn} under h is pseudo-Cauchy, then {cn} is pseudo- Cauchy with the same metric. Proof. Let {dn} be a rearrangement sequence of {cn} under h. Suppose {dn} ∈ P(σ). Let ε > 0 be given and m ∈ N where m = h(n);n ∈ N. Since n ∈ N and by assumption, there exist k, l ∈ N, k ̸= l such that k > n; l > n and σ(dk, dl) < ε. Since k, l ∈ N, k ̸= l we have h(k), h(l) ∈ N, h(k) ̸= h(l), by h is one-one. Since h is strictly increasing and k > n; l > n we have h(k) > h(n);h(l) > h(n). Now σ(ch(k), ch(l)) = σ(dk, dl) < ε. Thus, there exist h(k), h(l) ∈ N, h(k) ̸= h(l) such that h(k) > h(n) = m;h(l) > h(n) = m and σ(ch(k), ch(l)) < ε. Hence {cn} is pseudo-Cauchy with the same metric. 4. Asymptotic sequences Here, in a generalized metric space, two kinds of new sequences are defined and explore their nature. We begin with a definition of asymptotic sequence in a generalized metric space. Definition 26. Let (X,Ω) be a generalized metric space. A pair of sequence {sn}n∈N and {tn}n∈N in X is said be: (a) asymptotic with respect to σ ∈ Ω if for every ε > 0, there exists n0 ∈ N such that σ(sn, tn) < ε for all n ≥ n0. (b) uniformly asymptotic with respect to σ ∈ Ω if for every ε > 0, there exists n0 ∈ N such that σ(sm, tn) < ε for all m,n ≥ n0. Theorem 27 gives new tricks to check whether, in a generalized metric space, a given pair of sequence is in C(σ) or not. Theorem 27. Let (X,Ω) be a GMS. If the pair of sequence {cn}n∈N and {dn}n∈N is asymptotic with respect to σ ∈ Ω, then the followings are true. (a) If {dn} ∈ C(σ), then {cn} ∈ C(σ). (b) If {dn} is a convergent sequence with respect to σ ∈ Ω, then {cn} is a convergent sequence with the same metric. Proof. Assume that, {cn}n∈N and {dn}n∈N is asymptotic with respect to σ ∈ Ω. (a) Given that {dn} ∈ C(σ). Let ε > 0. Then there is N0 ∈ N such that σ(dn, dm) < ε 3 for all n,m ≥ N0. By hypothesis, there exists a positive integer N1 such that σ(cn, dn) < ε 3 V. Subramanian et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1869-1886 1877 for all n ≥ N1. Take N = max{N0, N1 + 1}. Then for n,m ≥ N, σ(cn, cm) ≤ σ(cn, dn) + σ(dn, dm) + σ(dm, cm) and so σ(cn, cm) < ε for all n,m ≥ N. Hence {cn} ∈ C(σ). (b) Suppose that {dn} is a convergent sequence with respect to σ ∈ Ω. Let ε > 0 be given and y ∈ X be a limit point of {dn}. Then there is n0 ∈ N such that σ(dn, y) < ε 2 for all n ≥ n0. By hypothesis, there is n1 ∈ N such that σ(cn, dn) < ε 2 for all n > n1. Take M0 = max{n0, n1 + 1}. Then σ(cn, y) ≤ σ(cn, dn) + σ(dn, y) and so σ(cn, y) < ε for all n ≥ M0. Thus, {cn} is convergent to y with respect to σ. Hence {cn} is a convergent sequence with respect to σ ∈ Ω. Theorem 28. Let (X,Ω) be a GMS, the pair of sequence {cn}n∈N and {dn}n∈N be asymp- totic with respect to the metric σ ∈ Ω. If {dn} ∈ P(σ), then {cn} ∈ P(σ). Proof. Assume that, {cn} and {dn} is asymptotic with respect to the metric σ ∈ Ω and {dn} ∈ P(σ). Let ε > 0 and m ∈ N. Then there exists N1 ∈ N such that σ(cn, dn) < ε 3 for all n ≥ N1 and there exist k1, j1 ∈ N, k1 ̸= j1 such that k1, j1 > m and σ(dk1 , dj1) < ε 3 . Case 1: Suppose that, N1 < m. Take k = k1, j = j1. Then k ̸= j and k, j > m. Now σ(ck, cj) ≤ σ(ck, dk) + σ(dk, dj) + σ(dj , cj) < ε 3 + ε 3 + ε 3 = ε. Thus, σ(ck, cj) < ε. Case 2: Assume that, N1 > m. If k1, j1 > N1, then we take k = k1, j = j1. By same argument in Case 1, we get σ(ck, cj) < ε. Suppose that, k1, j1 < N1. Since {dn} ∈ P(σ) and N1 ∈ N, there is k2, j2 ∈ N, k2 ̸= j2 such that k2, j2 > N1 and σ(dk2 , dj2) < ε 3 . Take k = k2, j = j2. Then σ(ck, cj) ≤ σ(ck, dk) + σ(dk, dj) + σ(dj , cj) and so σ(ck, cj) < ε. If k1 < N1 < j1, then there is k2, j2 ∈ N, k2 ̸= j2 such that k2, j2 > j1 and σ(dk2 , dj2) < ε 3 , since {dn} is a pseudo-Cauchy sequence and j1 ∈ N. Take k = k2, j = j2. Then σ(ck, cj) ≤ σ(ck, dk) + σ(dk, dj) + σ(dj , cj) and so σ(ck, cj) < ε. From all the cases, we get for every ε > 0 and m ∈ N, there exist k, j ∈ N, k ̸= j such that k, j > m and σ(xk, xj) < ε. Therefore, {cn} ∈ P(σ). Theorem 29 provides an easier way to check the nature of a given pair of sequence using the tool namely, asymptotic. Theorem 29. Let (X,Ω) be a generalized metric space, the pair of sequence {cn}n∈N and {dn}n∈N be asymptotic with respect to the metric σ ∈ Ω. If {dn} ∈ B(σ), then {cn} ∈ B(σ). Proof. Let ε > 0 be given. Then there exists N1 ∈ N such that σ(cn, dn) < ε for all n ≥ N1 and there exist m1, n1 ∈ N such that whenever n > j ≥ n1, the points dj and dn can be joined by an ε-chain of length m1. Case 1: If N1 > n1, then take k = m1+2 and l = N1. Then k, l ∈ N. If n > i ≥ l, then we can choose elements between ci and cn, namely, ci, di, di+1, ..., dn−1, dn, cn. Since n1 ≤ i < n, the points di and dn can be joined by an ε-chain of length m1, by assumption. Thus, σ(di, di+1) < ε, σ(di+1, di+2) < ε, ..., σ(dn−1, dn) < ε. Also, σ(cn, dn) < ε and σ(ci, di) < ε. Therefore, the points ci and cn can be joined by an ε-chain of length k. Case 2: Suppose that N1 < n1. Choose k = m1 + 2 and l = n1. Then by same argument as in Case 1, we get the points ci, cn such that ci and cn can be joined by an ε-chain of length k. Hence in both cases, we get that the sequence {cn} ∈ B(σ). V. Subramanian et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1869-1886 1878 The following Theorem 30 is reduce the complexity for finding whether, in a generalized metric space, a given pair of sequence is uniformly asymptotic or not using asymptotic. Theorem 30. Let (X,Ω) be a generalized metric space, the pair of sequence {cn}n∈N and {dn}n∈N be asymptotic with respect to σ ∈ Ω. If either {cn} or {dn} is Cauchy sequence with respect to σ ∈ Ω, then the pair of sequence {cn} and {dn} is uniformly asymptotic with respect to σ. Proof. Given that the pair of sequence {cn} and {dn} is asymptotic with respect to σ ∈ Ω. Let ε > 0. Then there is N1 ∈ N such that σ(cn, dn) < ε 2 for all n ≥ N1. If {cn} ∈ C(σ), then there is N0 ∈ N such that σ(cn, cm) < ε 2 for all n,m ≥ N0. Take N = max{N0, N1 + 1}. Then for n,m ≥ N, σ(cm, dn) ≤ σ(cm, cn) + σ(cn, dn) and so σ(cm, dn) < ε for all n,m ≥ N. Hence the pair of sequence {cn} and {dn} is uniformly asymptotic with respect to σ. Similarly, we can prove that the pair of sequence {cn} and {dn} is uniformly asymptotic if {dn} ∈ C(σ). Theorem 31 provides the necessary condition for a given pair of a convergent sequence is asymptotic in a generalized metric space. Theorem 31. Let (X,Ω) be a generalized metric space and {cn}, {dn} be convergent sequences with respect to σ ∈ Ω. If they have the limit points joined by an ε-chain of length 1 for every ε > 0, then the pair of sequence is asymptotic with the same metric. Proof. Let c, d ∈ X be the limit points of the sequences {cn} and {dn}, respectively with respect to σ. Let ε > 0 be given. Then there exists N0, N1 ∈ N such that σ(cn, c) < ε 3 for every n ≥ N0 and σ(dn, d) < ε 3 for every n ≥ N1. Take M = max{N0, N1}. For n ≥ M,σ(cn, dn) ≤ σ(cn, c) + σ(c, d) + σ(y, dn) and so σ(cn, dn) < ε 3 + σ(c, d) + ε 3 . By assumption, the points c and d can be joined by an ε 3 -chain of length 1. Thus, σ(c, d) < ε 3 and hence σ(cn, dn) < ε for n ≥ M. Hence the pair of sequence {cn} and {dn} is asymptotic with respect to the metric σ. The following Example 32 shows that the necessary condition in Theorem 31 can not be dropped. Example 32. Consider the generalized metric space (X,Ω1) where X = N ∪ {0},Ω1 = {σ1, σ2, σ3} ⊂ ΩX and the metrics are defined by σ1(c, d) = |c−d|;σ2(c, d) = min{σ1(c, d), 1} and σ3(c, d) = σ1(c,d) 1+σ1(c,d) for all c, d ∈ X. Define the sequence {cn} and {dn} by {cn} = { 1 n} and {dn} = {1 + 1 n}. Then {cn} converges to 0 and {dn} converges to 1 with respect to σ1. Here the points 0 and 1 can not be joined by an ε-chain of length m where ε > 0,m is a positive integer. Also, for every ε > 0, there is no n0 ∈ N such that σ1(cn, dn) < ε for all n > n0 Theorem 33. Let (X,Ω) be a generalized metric space. Then every convergent sequence and its convergent subsequence are asymptotic with the same metric. Proof. Let {cn} be a convergent sequence with respect to the metric σ ∈ Ω and {zn} be a convergent subsequence of {cn} with the metric σ. Let x be a limit point of the sequence V. Subramanian et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1869-1886 1879 {cn}. Then x is also a limit point of the subsequence {zn}. Let ε > 0 be given. Then there exists N0, N1 ∈ N such that σ(cn, x) < ε 2 for all n ≥ N0 and σ(zn, x) < ε 2 for all n ≥ N1. Choose M = max{N0, N1}. For n ≥ M,σ(cn, zn) ≤ σ(cn, x) + σ(x, zn) < ε 2 + ε 2 = ε and so σ(cn, zn) < ε for n ≥ M. Hence the pair of sequence {cn} and {zn} is asymptotic with respect to the metric σ. In a generalized metric space, the existence of a convergent sequence using convergent subsequence can be found straightforwardly by Theorem 34. Theorem 34. Let (X,Ω) be a generalized metric space and {cn}n∈N has a convergent subsequence {zn}n∈N with respect to σ ∈ Ω. If the pair of sequence {cn} and {zn} is asymptotic with respect to the metric σ, then {cn} is a convergent sequence with the same metric. Proof. Given that {zn} is a convergent subsequence of {cn} and z is a limit point of the sequence {zn} with respect to σ ∈ Ω. Let ε > 0 be given. Then there exists N0 ∈ N such that σ(zn, z) < ε 2 for every n ≥ N0. Since the pair of sequence {cn} and {zn} is asymptotic with respect to σ, there exists N1 ∈ N such that σ(cn, zn) < ε 2 for every n ≥ N1. Take M = max{N0, N1 + 1}. For n ≥ M, σ(cn, z) ≤ σ(cn, zn) + σ(zn, z) < ε 2 + ε 2 = ε. Thus, σ(cn, z) < ε for every n ≥ M. Hence {cn} is a convergent sequence with respect to σ. The condition that convergence on the sequence {cn} can not be dropped in Theorem 33 as shown by the following Example 35. Also, it shows that the condition asymptotic is necessary in Theorem 34. Example 35. Consider the generalized metric space (X,Ω1) whereX = R,Ω1 = {σ1, σ2, σ3} ⊂ ΩX and the metrics are defined by σ1(c, d) = |c−d| 1+|c−d| ;σ2(c, d) = min{σ1(c, d), 1} and σ3(c, d) = { 0 if c = d, 1 if c ̸= d for all c, d ∈ X. Let {cn} = {(−1)n}n∈N be a sequence in X and {dn} be a subsequence of {cn} defined by dn = 1 for all n ∈ N. Then {dn} is a convergent subsequence of {cn} with respect to the metric σ1 ∈ Ω1. Here {cn} is not a convergent sequence. Also, {cn} and {dn} are not asymptotic with respect to the metric σ1. The following Example 36 shows that the condition convergent on {zn} can not be dropped in Theorem 34. Example 36. Consider the generalized metric space (X,Ω1) whereX = R,Ω1 = {σ1, σ2, σ3} ⊂ ΩX and the metrics are defined by σ1(c, d) = |c− d|;σ2(c, d) = min{σ1(c, d), 1} and σ3(c, d) = { 0 if c = d, 1 if c ̸= d for all c, d ∈ X. Let V. Subramanian et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1869-1886 1880 {cn} = { 1 if n is odd 2 if n is even be a sequence in X and {zn} be a subsequence of {cn} defined by zn = cn+2 for all n ∈ N. Then the pair of sequence {cn} and {zn} is asymptotic with respect to the metric σ1. But {cn} is not a convergent sequence with respect to the metric σ1. Because, {zn} is not a convergent subsequence of {cn} with respect to the metric σ1 ∈ Ω1. {dn} ∈ B(σ) {cn} ∈ C(σ) {dn} ∈ P(σ) {dn} ∈ C(σ) The following Theorem 37 describe the above diagram. Theorem 37. Let (X,Ω) be a generalized metric space, the pair of sequence {cn} and {dn} where cn ̸= dn for all n ∈ N be uniformly asymptotic with respect to σ ∈ Ω. Then {cn} ∈ C(σ) if any one of the following hold. (a) {dn} ∈ B(σ). (b) {dn} ∈ C(σ). (c) {dn} ∈ P(σ). Proof. Let ε > 0 be given. Then there exists N0 ∈ N such that σ(cm, dn) < ε 3 for every m,n ≥ N0, since the pair of sequence {cn} and {dn} is uniformly asymptotic. (a) Given that {dn} ∈ B(σ). Then there exist l, k0 ∈ N such that whenever n > j ≥ k0, the points dj and dn are joined by an ε 3 -chain of length l. Choose M = max{N0 + 1, k0}. For n,m ≥ M, σ(cn, cm) ≤ σ(cn, dn+1) + σ(dn+1, dn+2) + σ(dn+2, cm) < ε 3 + ε 3 + ε 3 = ε. Thus, σ(cn, cm) < ε for every n,m ≥ M. Therefore, {cn} ∈ C(σ). (b) Let {dn} ∈ C(σ). Then there exists an infinite subset Nε of N such that σ(dk, dj) < ε 3 for all k, j ∈ Nε. Let Nk = min{Ni : Ni ∈ Nε} and take M = max{N0 + 1, Nk}. For n,m ≥ M,σ(cn, cm) ≤ σ(cn, dj) + σ(dj , dk) + σ(dk, cm), since Nε is an infinite subset of N, there exist j, k ∈ Nε such that j, k > M. Thus, σ(cn, cm) < ε 3 + ε 3 + ε 3 = ε. Hence σ(cn, cm) < ε for all n,m ≥ M. Therefore, {cn} ∈ C(σ). (c) Suppose that {dn} ∈ P(σ). Then there exists k, l ∈ N with k ̸= l and k, l > N0 such that σ(dk, dl) < ε 3 . TakeM = N0+1. For n,m ≥ M,σ(cn, cm) ≤ σ(cn, dk)+σ(dk, dl)+σ(dk, cm). Thus, σ(cn, cm) < ε 3 + ε 3 + ε 3 = ε. Therefore, σ(cn, cm) < ε for all n,m ≥ M. Hence {cn} ∈ C(σ). Theorem 38. Let (X,Ω) be a GMS. Then (X,Ω) is a weakly complete space if and only if there is a kernel Ω0 ⊂ Ω and if the pair of sequence {cn} and {dn} is asymptotic with respect to σ ∈ Ω0 and {dn} is Cauchy with respect to σ, then {cn} is a convergent sequence with the same metric. Proof. Given that (X,Ω) is a weakly complete space. Then there exists a kernel Ω0 ⊂ Ω consisting of complete metrics. Let the pair of sequence {cn} and {dn} be asymptotic with V. Subramanian et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1869-1886 1881 respect to σ ∈ Ω0 and {dn} is Cauchy with the same metric. Then {dn} is convergent with respect to σ. By Theorem 27 (b), {cn} is a convergent sequence with the same metric. Conversely, let {zn} be a Cauchy sequence with respect to the metric σ ∈ Ω0. Define the sequence {tn} in X by t1 = 2, t2 = 1 and tn = zn−2 for n ≥ 3. Since {zn} is Cauchy, for every ε > 0, there exists N0 ∈ N such that σ(zn, tn) < ε for every n ≥ N0. Thus, the pair of sequence {zn} and {tn} is asymptotic with respect to σ ∈ Ω0. By assumption, {tn} is a convergent sequence with the same metric. By Theorem 27 (b), {zn} is a convergent sequence with respect to σ ∈ Ω0. Since σ is arbitrary, Ω0 consisting of complete metrics. Therefore, (X,Ω) is a weakly complete space. Definition 39. Let (X,Ω1) and (Y,Ω2) be two generalized metric spaces. A function h : (X,Ω1) → (Y,Ω2) is said to be Cauchy-continuous if {h(zn)}n∈N is a Cauchy sequence in (Y,Ω2) for any Cauchy sequence {zn}n∈N in (X,Ω1). That is, h is Cauchy-continuous if {zn} is Cauchy with respect to σ ∈ Ω1, then there is σ ∈ Ω2 such that {h(zn)} is Cauchy with respect to σ. Theorem 40. Let (X,Ω1) and (Y,Ω2) be two GMSs. If h : X → Y is a Cauchy continuous map and if {cn} is a cofinally-Cauchy sequence in X, then {h(cn)} has a cofinally-Cauchy subsequence in Y. Proof. Let {cn} be a cofinally-Cauchy sequence in X. Then there exists a metric σ ∈ Ω1 such that {cn} is a cofinally-Cauchy sequence with respect to this metric in X. By Theorem 19, {cn} has a Cauchy subsequence {cnk } with respect to the same metric in X. Since h is Cauchy continuous, there exists a metric d ∈ Ω2 such that {h(cnk )} is Cauchy with respect to d. By Theorem 13 (a), {h(cnk )} is a cofinally-Cauchy sequence with respect to the metric d. Hence {h(cn)} has a cofinally-Cauchy subsequence in Y. Definition 41. Let (X,Ω1) and (Y,Ω2) be two generalized metric spaces. A function h : X → Y is said to be: i). Bourbaki-Cauchy regular (in short, BC-regular) if {h(zn)}n∈N is a Bourbaki Cauchy sequence in (Y,Ω2) whenever {zn}n∈N is Bourbaki Cauchy in (X,Ω1). ii). cofinally-Cauchy regular (briefly, CC-regular) if {h(zn)}n∈N is a cofinally Cauchy sequence in (Y,Ω2) whenever {zn}n∈N is cofinally Cauchy in (X,Ω1). iii). pseudo-Cauchy regular (in short, PC-regular) if {h(zn)}n∈N is a pseudo Cauchy sequence in (Y,Ω2) whenever {zn}n∈N is pseudo Cauchy in (X,Ω1). That is, h is Bourbaki-Cauchy regular (resp. CC-regular, PC-regular) if {zn}n∈N is Bourbaki (resp. cofinally, pseudo) Cauchy with respect to σ1 ∈ Ω1, then there is d ∈ Ω2 such that {h(zn)}n∈N is Bourbaki (resp. cofinally, pseudo) Cauchy with respect to d. The following three theorems (Theorem 42, Theorem 43 and Theorem 44) are give some relations between the functions defined in above Definition 41. V. Subramanian et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1869-1886 1882 Theorem 42. Let (X,Ω1) and (Y,Ω2) be two GMSs, h : X → Y be a BC-regular map. If every cofinally-Cauchy sequence {cn} in X, there exists a sequence {dn} in X with cn ̸= dn for every n ∈ N such that the pair of sequence {cn} and {dn} is uniformly asymptotic with respect to the same metric, then h is a CC-regular map. Proof. Let {an} ∈ C(σ) where σ ∈ Ω1. Then by hypothesis, there exists a sequence {bn} in X with an ̸= bn for every n ∈ N such that the pair of sequence {an} and {bn} is uniformly asymptotic with respect to σ. By Theorem 37, {bn} ∈ C(σ) and so {bn} ∈ B(σ), by Theorem 11(a). Then {an} ∈ B(σ), by Theorem 29. Since h is a BC-regular map, there is a metric d ∈ Ω2 such that {h(an)} ∈ B(d) where d ∈ Ω2. By Theorem 17, {h(an)} ∈ C(d) where d ∈ Ω2. Hence h is a CC-regular map. Theorem 43. Let (X,Ω1) and (Y,Ω2) be two generalized metric spaces, h : X → Y be a Cauchy-continuous map. If for every Bourbaki-Cauchy sequence {cn} in X, there exists a sequence {dn} in X with cn ̸= dn for every n ∈ N such that the pair of sequence {cn} and {dn} is uniformly asymptotic with the same metric, then h is a BC-regular map. Proof. Let {an} ∈ B(σ) where σ ∈ Ω1. Then by hypothesis, there exists a sequence {bn} in X with an ̸= bn for every n ∈ N such that the pair of sequence {an} and {bn} is uniformly asymptotic with the same metric. By Theorem 37, {bn} ∈ C(σ). Then {an} ∈ C(σ), by Theorem 27. Since h is a Cauchy-continuous map, there is a metric d ∈ Ω2 such that {h(an)} ∈ C(d) where d ∈ Ω2. By Theorem 11(a), {h(an)} ∈ B(d) where d ∈ Ω2. Hence h is a BC-regular map. Theorem 44. Let (X,Ω1) and (Y,Ω2) be two generalized metric spaces, h : X → Y be a CC-regular map. If for every cofinally-Cauchy sequence {sn} in Y, there exists a sequence {tn} in Y with sn ̸= tn for every n ∈ N such that the pair of sequence {sn} and {tn} is uniformly asymptotic with the same metric, then h is a Cauchy-continuous map. Proof. Let {an} ∈ C(σ) for σ ∈ Ω1. By Theorem 13, {an} ∈ C(σ).. Since h is a CC-regular map, there is a metric d ∈ Ω2 such that {h(an)} ∈ C(d) where d ∈ Ω2. By hypothesis, there exists a sequence {bn} in Y with h(an) ̸= bn for every n ∈ N such that the pair of sequence {h(an)} and {bn} is uniformly asymptotic with the same metric. By Theorem 37, {bn} ∈ C(d) for d ∈ Ω2. Then {h(an)} ∈ C(d) where d ∈ Ω2, by Theorem 27. Hence h is a Cauchy-continuous map. The following Theorem 45 gives a necessary condition for a weakly complete metric space to be a cofinally-complete metric space. Theorem 45. Let (X,Ω) be a weakly complete space. If for every cofinally-Cauchy se- quence {sn} in X, there exists a sequence {tn} in X with sn ̸= tn for every n ∈ N such that the pair of sequence {sn} and {tn} is uniformly asymptotic with the same metric, then (X,Ω) is a cofinally-complete space. Proof. Given that (X,Ω) is a weakly complete space. Then there exists a kernel Ω0 ⊂ Ω consisting of complete metrics. Let σ0 ∈ Ω0 and {an} ∈ C(σ0). By hypothesis, there exists a sequence {tn} in X with an ̸= tn for every n ∈ N such that the pair of sequence {an} V. Subramanian et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1869-1886 1883 and {tn} is uniformly asymptotic with the same metric. By Theorem 37, {tn} ∈ C(σ0) and hence it is a convergent sequence, by assumption. Hence {an} is a convergent sequence with respect to σ0, by Theorem 27(b). Thus, Ω0 consisting of complete metrics. Therefore, (X,Ω) is a cofinally-complete space. In Theorem 45, if we replace “cofinally-Cauchy” by “pseudo-Cauchy” we get that (X,Ω) is a pseudo-complete space. Theorem 46. Let (X,Ω) be a GMS, the pair of sequence {cn} and {dn} is asymptotic, {dn} ∈ B(σ). If every ε k -chain have length k − 1 where k ∈ N, ε > 0, then the pair of sequence {cn} and {dn} is uniformly asymptotic with respect to the same metric. Proof. Let ε > 0 be given and m ∈ N. Since ε m > 0 and {dn} ∈ B(σ), there exist n1,m1 ∈ N whenever n > j ≥ n1 such that the points dj and dn can be joined by an ε m -chain of length m1. By hypothesis, m1 = m− 1. Since ε m > 0 and the pair of sequence {cn}, {dn} is asymptotic, there exists N0 ∈ N such that σ(cn, dn) < ε m for all n ≥ N0. Take M = max{n1, N0}. For n,m ≥ M, Case 1: If m < n, then M ≥ m < n. By assumption, the points dm and dn can be joined by an ε m -chain of length m− 1. Thus, σ(dm, dn) < (m−1)ε m . Now σ(cm, dn) ≤ σ(cm, dm) + σ(dm, dn) < ε m + (m−1)ε m = ε. Case 2: Suppose that m > n. Then by similar argument as in Case-1, we get σ(cm, dn) < ε. From Case 1 and Case 2, we get, σ(cm, dn) < ε for all m,n ≥ M. Hence the pair of sequence {cn} and {dn} is uniformly asymptotic with respect to the metric σ ∈ Ω. Next, the rest of this section, in a hyperconnected space, some interesting results for uniformly asymptotic sequence are proven. Theorem 47. Let (X,Ω) be a GMS and {cn}, {dn} be Cauchy, cofinally-Cauchy sequences with respect to σ ∈ Ω in X, respectively. If (X,µΩ) is a hyperconnected space, then the pair of sequence {cn} and {dn} is uniformly asymptotic with the same metric. Proof. Let ε > 0 be given. Then there is n0 ∈ N such that σ(cn, cm) < ε 4 for all n,m ≥ n0 and there is an infinite subset Nε of N such that σ(dk, dl) < ε 4 for every k, l ∈ Nε. Take M0 = max{n0, n1 | n1 = inf{Nε}}. If Bσ(cn, ε 4) ∩ Bσ(dl, ε 4) = ∅, then Bσ(cn, ε 4) ⊂ X−Bσ(dl, ε 4). This implies iµΩ(X−Bσ(dl, ε 4)) ̸= ∅ which implies that cµΩ(Bσ(dl, ε 4)) ̸= X. But Bσ(dl, ε 4) is µΩ-dense, since (X,µΩ) is a hyperconnected space. Therefore, Bσ(cn, ε 4)∩ Bσ(dl, ε 4) ̸= ∅. Let t ∈ Bσ(cn, ε 4) ∩ Bσ(dl, ε 4). Then σ(cn, t) < ε 4 and σ(dl, t) < ε 4 . For l1,m1 ≥ M0, σ(cl1 , dm1) ≤ σ(cl1 , cn) + σ(cn, t) + σ(t, dl) + σ(dl, dm1). If m1 ∈ Nε, then we get σ(dl, dm1) < ε 4 . Therefore, σ(cl1 , dm1) < ε for all l1,m1 ≥ M0. Assume that, m1 /∈ Nε. Since (X,µΩ) is a hyperconnected space, Bσ(dl, ε 8) is µΩ-dense. Then Bσ(dl, ε 8)∩ Bσ(dm1 , ε 8) ̸= ∅, since Bσ(dm1 , ε 8) ∈ µ̃Ω. Let z ∈ Bσ(dl, ε 8)∩Bσ(dm1 , ε 8). Then σ(dl, dm1) ≤ σ(dl, z)+σ(z, dm1) < ε 8 + ε 8 = ε 4 . Therefore, σ(cl1 , dm1) < ε for all l1,m1 ≥ M0. Hence the pair of sequence {cn} and {dn} is uniformly asymptotic sequence with respect to σ ∈ Ω in X. V. Subramanian et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1869-1886 1884 Corollary 48. Let (X,Ω) be a GMS and {cn}, {dn} be Cauchy sequences with respect to σ ∈ Ω in X. If (X,µΩ) is a hyperconnected space, then the pair of sequence {cn} and {dn} is uniformly asymptotic sequence with respect to the same metric. Theorem 49. Let (X,Ω) be a GMS and {cn}, {dn} be Cauchy, convergent sequences with respect to σ ∈ Ω in X, respectively. If (X,µΩ) is a hyperconnected space, then {cn} is a convergent sequence with respect to σ. Proof. Let ε > 0 be given and y be a limit point of {dn}. Then there exist positive integers n0, n1 ∈ N such that σ(cn, cm) < ε 2 for all n,m ≥ n0 and σ(dn, y) < ε 2 for all n ≥ n1. Take M = max{n0, n1}. Fix n ≥ M. Since (X,µΩ) is a hyperconnected space, Bσ(cn, ε 2)∩Bσ(y, ε 2) ̸= ∅. Let t ∈ Bσ(cn, ε 2)∩Bσ(y, ε 2). Then σ(cn, y) ≤ σ(cn, t)+σ(t, y) < ε 2 + ε 2 = ε. Therefore, σ(cn, y) < ε for all n ≥ M. Hence {cn} is a convergent sequence with respect to σ. Definition 50. Let (X,Ω) be a GMS. Then Ω is said to satisfy the S-property if σi(c, d) ≤ σj(c, d) or σj(c, d) ≤ σi(c, d) for any σi, σj ∈ Ω and c, d ∈ X. Theorem 51. Let (X,Ω) be a GMS, Ω satisfies the S-property and the pair of sequences {cn} and {dn} be asymptotic with respect to σj ∈ Ω. Then the following hold. (a) If {cn} ∈ C(σi), then {dn} ∈ C(σi). (b) If {cn} is convergent with respect to σi ∈ Ω in X, then {dn} is convergent with respect to σi. Proof. (a). Suppose that {cn} ∈ C(σi). Let ε > 0 be given. Then there is positive integers n0, n1 such that σj(cn, dn) < ε 3 for all n > n0 and σi(cn, cm) < ε 3 for all n,m ≥ n1. By hypothesis, σi(c, d) ≤ σj(c, d) or σj(c, d) ≤ σi(c, d). Case 1: If σi(c, d) < σj(c, d), then σi(cn, dn) < ε 3 for all n > n0. Choose N0 = max{n0, n1}. For n,m > N0, σi(dn, dm) ≤ σi(dn, cn)+σi(cn, cm)+σi(cm, dm) < ε 3+ ε 3+ ε 3 = ε. Therefore, σi(dn, dm) < ε for all n,m > N0. Hence {dn} ∈ C(σi). Case 2: If σj(c, d) < σi(c, d), then the proof is completed by replacing i by j as in the proof of Case-1. (b). It is obvious. Theorem 52. Let (X,Ω) be a GMS, Ω satisfy the S-property and the pair of sequences {cn} and {dn} be uniformly asymptotic with respect to σj ∈ Ω. Then the following hold. (a) If {dn} ∈ C(σi), then {dn} ∈ C(σi). (b) If {dn} ∈ P(σi), then {dn} ∈ C(σi). In [14], Preecha Yupapin et. al have obtained new set of properties of nowhere dense sets. Inspired by this, rest of this section, we check some of the collections are whether satisfies the stack property or not, in a generalized metric space. Let (X,µ) be a GTS. A family E of subsets of X is called a stack [13] if C ∈ E whenever D ∈ E and D ⊂ C. Theorem 53. Let (X,Ω) be a GMS and η = {{cn}n∈N | {cn}n∈N is a cofinally-Cauchy sequence in X}. Then η is a stack. REFERENCES 1885 Proof. Let {dn}n∈N be a subsequence of {cn}n∈N in X where {dn}n∈N ∈ η. Then there is a metric σ ∈ Ω such that {dn}n∈N ∈ C(σ) and so for every ε > 0, there exists an infinite subset Nε of N such that for every n, j ∈ Nε we have σ(dn, dj) < ε. By hypothesis, σ(cn, cj) < ε for every n, j ∈ Nε. Thus, for every ε > 0, there exists an infinite subset Nε of N such that for every n, j ∈ Nε we have σ(cn, cj) < ε. Therefore, {cn}n∈N ∈ C(σ). Hence {cn}n∈N ∈ η. Thus, η is a stack in (X,Ω). Theorem 54. Let (X,Ω) be a generalized metric space and η = {{cn}n∈N | {cn}n∈N is a pseudo-Cauchy sequence in X}. Then η is a stack. Proof. Let {ln}n∈N be a subsequence of {kn}n∈N in X where {ln}n∈N ∈ η. Then there is a metric σ ∈ Ω such that {ln}n∈N ∈ P(σ) and so for every ε > 0 and for every n ∈ N, there exist s, t ∈ N, s ̸= t such that s, t > n and σ(ls, lt) < ε. By hypothesis, σ(ks, kt) < ε. Thus, for every ε > 0 and for every n ∈ N, there exist s, t ∈ N, s ̸= t such that s, t > n and σ(ks, kt) < ε. Therefore, {kn}n∈N ∈ P(σ). Hence {kn}n∈N ∈ η. Thus, η is a stack in (X,Ω). Theorem 55. Let (X,Ω) be a generalized metric space. If η = {Ω0 | Ω0 is a kernel in X}, then η is a stack. Proof. Let Ω1 ⊂ Ω0 where Ω1 ∈ η. Then Ω1 is a kernel. Let K ∈ µ̃Ω. 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