EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 3, 2023, 1381-1388 ISSN 1307-5543 – ejpam.com Published by New York Business Global Hesitant Fuzzy Compactness and Hesitant Fuzzy Regularity in Hesitant Fuzzy Topological Spaces A. Swaminathan1, Cenap Ozel2, Ibtesam Alshammari3,∗ 1 Department of Mathematics, Government Arts College(A), Kumbakonam, Tamil Nadu- 612 002, India 2 Department of Mathematics, King Abdulaziz University, Jeddah-21589, Saudi Arabia 3 Department of Mathematics, University of Hafr Al-Batin, Hafr Al-Batin, Saudi Arabia Abstract. We define hesitant fuzzy maximal open cover to establish hesitant fuzzy m-compactness and discuss its properties. Further we obtain few more results on hesitant fuzzy minimal c-regular and minimal c-normal spaces. We have proved that a hesitant fuzzy Haussdorff m-compact space is hesitant fuzzy minimal c-normal. 2020 Mathematics Subject Classifications: 54A40, 03E72 Key Words and Phrases: Hesitant fuzzy minimal open, hesitant fuzzy maximal open cover, hesitant fuzzy m-compact, hesitant fuzzy minimal c-regular 1. Introduction Origination of fuzzy sets by Zadeh[11] emerged many branches of mathematics for many decades. Chang[1] introduced fuzzy topology in 1968. As an addendum to fuzzy sets, the notion hesitant fuzzy set introduced by Torra[4] in 2010. Deepak et. al. [2] intro- duced hesitant fuzzy topological space and extended the study to hesitant connectedness and compactness in hesitant fuzzy topological space. The notions of hesitant fuzzy min- imal, maximal open[9] and hesitant fuzzy minimal, maximal clopen[7] sets introduced by Swaminathan and Sivaraja. Also the idea of hesitant fuzzy mean open and closed sets[8] investigated by Swaminathan and Sivaraja. In section 2, we define a new notion hesitant fuzzy maximal open cover in hesitant fuzzy topological space. Section 3 of this paper, the concept of hesitant fuzzy m-compact space and some properties are discussed. In section 4, the notion of hesitant fuzzy minimal c-regular (resp.c-normal spaces) are extended from which it is showed that a hesitant fuzzy Haussdorff m-compact space is hesitant fuzzy minimal c-normal. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4768 Email addresses: asnathanway@gmail.com (A. Swaminathan), cenap.ozel@gmail.com (C. Ozel), iealshamri@uhb.edu.sa, iealshamri@hotmail.com (I. Alshammari) https://www.ejpam.com 1381 © 2023 EJPAM All rights reserved. A. Swaminathan, Cenap Ozel, Ibtesam Alshammari / Eur. J. Pure Appl. Math, 16 (3) (2023), 1381-1388 1382 The following terminologies, “hesitant fuzzy minimal open set, hesitant fuzzy maximal open set, hesitant fuzzy mean open set,hesitant fuzzy clopen set, hesitant fuzzy cut-point space, hesitant fuzzy connected topological space,hesitant fuzzy disconnected topologi- cal space and hesitant fuzzy topological space”are respectively abbreviated as “HFMIO, HFMAO, HFMEO, HFCLO, HFCS, HFCTS,HFDTS and HFTS.” 2. Preliminaries Definition 2.1. [4] A HFS h in X is a function h : X → P [0, 1], where P [0, 1] represents the power set of [0, 1]. We define the hesitant fuzzy empty set h0(resp. whole set h1) is a HFS in X as follows: h0(x) = ϕ (resp. h1(x) = [0, 1]),∀x ∈ X. HS(X) stands for collection of HFS in X. Definition 2.2. [5] Let X be a nonempty set. A HFT τ of subsets X is said to be HFT on X if (i) h0, h1 ∈ τ . (ii) ⋃ i∈J hi ∈ τ for each (hi)i∈J ⊂ τ . (iii) h1 ∩ h2 ∈ τ for any h1, h2 ∈ τ . “The pair (X, τ) is called HFTS. The members of τ are called HFO sets in X. A HFS h in X is HFC set (in short HFC) in (X, τ) if hc ∈ τ .” Definition 2.3. [3] Two HFS h1 and h2 of X are said to be equal if h1 ⊂ h2 and h2 ⊂ h1. Definition 2.4. [4] Let h ∈ HS(X) for any nonempty set X. Then hc is the complement of h which is HFS in X such that hc(x) = [h(x)]c = [0, 1]\h(x). Definition 2.5. [5] Suppose that (X, τ) is a HFTS such that xλ ∈ Hp(X) and N ∈ HS(η). Then the hesitant fuzzy neighbourhood N of xλ is defined as if for an hesitant fuzzy set U ∈ τ such that xλ ∈ U ⊂ N . Definition 2.6. [9]A proper nonzero HFO set ξ of Xis said to be (i)HFMIO set if ξ and h0 are only HFO sets contained in ξ. (ii)HFMAO set if h1 and ξ are only HFO sets containing ξ. Definition 2.7. [9]A proper nonzero HFC set η of Xis said to be (i) HFMAC set if any HFC set which contains η is h1 or η. (ii)HFMIC set if any HFC set which is contained in η is h0 or η. Definition 2.8. [7] A proper HFCLO set φ of X is called a HFMICLO set if ϱ is a HFCLO set such that ϱ < φ, then ϱ = φ or ϱ = h0. Definition 2.9. [7] A proper HFCLO set φ of X is called a HFMACLO set if ϱ is a HFCLO set such that φ < ϱ, then φ = ϱ and ϱ = h1. Definition 2.10. [8] In a fts X, ξ is called a HFMEO(resp.γ FMEC) if ∃ λ, µ(̸= ξ) two distinct proper HFO sets (resp. two distinct proper hesitant fuzzy closed sets ζ, φ(̸= γ)) such that λ < ξ < µ(resp. ζ < γ < φ) A. Swaminathan, Cenap Ozel, Ibtesam Alshammari / Eur. J. Pure Appl. Math, 16 (3) (2023), 1381-1388 1383 Lemma 2.1.. [6] Each nonzero HFO set γ of a T1-fcts X is infinite and is not a HFMIO in X. Theorem 2.2.. [6] A proper HFO set γ of a T1-fcts X is a HFMEO set in X iff γ ̸= h1 − {xα} for any xα ∈ X. 2.1. Hesitant Fuzzy Maximal Open Cover and Hesitant Fuzzy m-compact Spaces We now introduce hesitant fuzzy maximal open covers. Further the idea of hesitant fuzzy m-compact space is studied by means of hesitant fuzzy maximal open covers. A hesitant fuzzy cover C of X is an hesitant fuzzy refinement of the hesitant fuzzy cover D of X if ∀ ξ ∈ C , ∃ζ ∈ D such that ξ < ζ. Definition 2.11. Let C and D be two hesitant fuzzy covers of a HFTS X.C is an hesitant fuzzy s-refinement of D if for each ξ ∈ C ∃ ζ ∈ D such that ξ < ζ. A hesitant fuzzy s- refinement C of D is said to be a HFO s-refinement of D if all members of C and D are HFO. It is clear that if D = {h1} and ξ ̸= h1 for each ξ ∈ C , then C is an hesitant fuzzy s-refinement of D . If C is hesitant fuzzy s-refinement of D then C is an hesitant fuzzy refinement of D . Further we see that no element of an s-hesitant fuzzy refinement of any hesitant fuzzy cover of X is HFMAO. Definition 2.12. A HFO cover C of a HFTS X is called a HFMAO cover of X if C is not an hesitant fuzzy s-refinement of any other HFO cover of X. Lemma 2.3.. A HFO cover containing a HFMAO set is hesitant fuzzy maximal. Proof. Obvious. Theorem 2.4. (Existence of HFMAO covers). There exists a HFMAO cover in an infinite T1-HFTS. Proof. LetX be an infinite T1-HFTS. Then for each xα ∈ X, h1−{xα} is HFMAO set in X. Let xβ ∈ X. Consider a finite hesitant fuzzy subset M = {xαi |xαi ̸= φ, i ∈ Z; 1 ≤ i ≤ n}. Also ξ in X is hesitant fuzzy closed as X is T1-HFTS. Henceforth {h1−{xβ}, h1−G} is HFO cover of X having HFMAO set h1−{xβ}. Hence by Lemma 2.3.,{h1−{xβ}, h1−M} is HFMAO cover of X. Theorem 2.5.. Any HFO cover M of an infinite T1-HFTS is a HFMAO cover of X iff M contains a HFMAO set. Proof. Let M = {Uk|k ∈ V } be a HFMAO cover of X such that no Uk,k ∈ V is HFMAO. By Theorem 2.2., Uk is not also HFMIO for each k ∈ V which implies that Uk,k ∈ V is HFMEO. So ∀ k ∈ V , ∃Vk a proper HFO set Vksuch that Uk < ̸ = Vk. Let A. Swaminathan, Cenap Ozel, Ibtesam Alshammari / Eur. J. Pure Appl. Math, 16 (3) (2023), 1381-1388 1384 N = {Vk|Uk ⪇ Vk, Uk ∈ M }. Clearly N is hesitant fuzzy cover of X. Therefore M is an hesitant fuzzy s-refinement of N a contradiction to tha fact that M is a HFMAO cover of X. Hence M has a HFMAO set as one among its members. The converse part follows by Lemma 2.3.. Definition 2.13. A HFTS X is said to be a hesitant fuzzy m-compact if each HFMO cover of X has a finite HFO s-refinement. Theorem 2.6.. Every infinite T1-HFCTS is hesitant fuzzy m-compact. Proof. Let M be HFMAO cover of an infinite T1-HFCTS X. By Theorem 2.5., M contains a HFMAO set U . By Theorem 2.5., take U = h1−{xα} for some xα ∈ X. There is an V ∈ M such that xα ∈ V . By Lemma 2.1., for hesitant fuzzy points xα, xβ ∈ V with xα ̸= xβ there are HFO sets V1 = h1 − {xα, xβ},V2 = V − {xα},V3 = V − {xβ} of X. Then {V1, V2, V3} is an hesitant fuzzy s-refinement of M . Example 2.7.. Let τ = {h0, h1, h1, h2, h3, h4} and (X, τ) be a hesitant fuzzy topological space where h1 = { [0, 1] if x ̸= 1 4 0 if x = 1 4 ;h2 = { 0 if x ̸= 1 4 [0, 1] if x = 1 4 ; h3 = { [0, 14 ] if x ̸= 1 4 [0, 1] if x = 1 4 ;h4 ={ [0, 14 ] if x ̸= 1 4 0 if x = 1 4 . Clearly (X, τ) is hesitant fuzzy compact but not hesitant fuzzy m-compact. Remark 2.8.. By Theorem 3.4, the real number space with the usual hesitant fuzzy topol- ogy is hesitant fuzzy m-compact but generally it is not hesitant fuzzy compact. Since by Theorem 2.6. along with Example 2.7., we conclude that both hesitant fuzzy compactness and hesitant fuzzy m-compactness are independent. Definition 2.14. A function f : X → Y is said to be hesitant fuzzy m-continuous if inverse image of each proper HFO set in Y is HFMAO in X. Theorem 2.9.. Let X be a hesitant fuzzy m-compact topological space and f : X → Y be a bijective hesitant fuzzy m-continuous function. Then Y is hesitant fuzzy m-compact. Proof. Let S(Y ) be a hesitant fuzzy cover of Y . Then S (X) = {f−1(Uk)|Uk ∈ S (Y )} is a HFMAO cover of X . By hesitant fuzzy m-compactness of X,S (X) has a finite hesitant fuzzy s-refinement S1 (X) = {f−1(Uk)|Uk ∈ S (Y ), k ∈ Z+} which gives S1 (Y ) = {f(f−1(Uk))|Uk ∈ S (Y ), k ∈ Z+} = {Uk|Uk ∈ S (Y ), k ∈ Z+} . For each k ∈ Z+, there exists U ∈ S (Y ) such that f−1(Uk) ⪇ f−1(U) gives Uk ⪇ U . Hence S1 (Y ) is a hesitant fuzzy finite s-refinement of S (Y ). Definition 2.15. A hesitant fuzzy point xα of a HFTS X is hesitant fuzzy m-complete accumulation point of any hesitant fuzzy subset M of X if |U∧M | = |M | for each HFMAO set U containing xα. A. Swaminathan, Cenap Ozel, Ibtesam Alshammari / Eur. J. Pure Appl. Math, 16 (3) (2023), 1381-1388 1385 Theorem 2.10.. Each infinite hesitant fuzzy subset of a hesitant fuzzy m-compact space has an hesitant fuzzy m-complete accumulation point. Proof. Let G be an infinite hesitant fuzzy subset of a hesitant fuzzy m-compact HFTS X. Assume for each xα ∈ X, there is a HFMAO set Wxα containing xα and satisfying |Wxα ∧ ϱ| < |ϱ|. Since {Wxα |xα ∈ X} is an HFO cover of X consists of HFMAO sets, by Lemma 2.3., {Wxα |xα ∈ X} is a HFMAO cover of X. Therefore a finite hesitant fuzzy s-refinement {Wxα |xαi ∈ X, i ∈ Z+} of {Wxα |xα ∈ X}. Now |ϱ| = | n ∨ i=1 (Wxα ∧ ϱ)| < |ϱ|, a contradiction. 2.2. Hesitant Fuzzy Minimal c-regular and Hesitant Fuzzy c-normal Spaces Definition 2.16. A HFTS X is called a hesitant fuzzy minimal c-regular if for each xα ∈ X and each HFMIC set γ with xα /∈ γ, there exists disjoint HFO sets λ,µ such that xα ∈ λ and λ < µ. Theorem 2.11.. Let X be a HFTS. Then the following are equivalent: (i) X is hesitant fuzzy minimal c-regular. (ii) Given a hesitant fuzzy point xα ∈ X and a HFMAO set ω containing xα , there is an HFO set ϱ such that xα ∈ ϱ < Cl(ϱ) < ω. (iii) For a hesitant fuzzy point xα ∈ X and a HFMIC set γ with xα /∈ γ, there exists HFO set ω containing xα such that Cl(ω) ∧ γ = h0. Proof. (i) ⇒ (ii), (ii) ⇒ (iii), (iii) ⇒ (i) : Proof follows. Definition 2.17. A HFTS X is called a hesitant fuzzy minimal c-normal if for each pair of distinct HFMIC sets η, γ there exists disjoint HFO sets λ,µ such that η < λ and γ < µ. Theorem 2.12.. Let X be a HFTS. Then the following are equivalent: (i) X is hesitant fuzzy minimal c-normal. (ii) For each HFMIC set ξ and each HFMAO set ω with ξ < ω ,there is a HFO set ϱ such that ξ < ϱ < Cl(ϱ) < ω. (iii) For each pair of distinct HFMIC sets ξ,ζ ,there exists disjoint HFO sets ω, ϱ disjoint HFO sets such that ξ < ω, Cl(ω) ∧ ζ = h0 and ζ < ϱ, Cl(ϱ) ∧ ξ = h0. (iv) For each pair of distinct HFMIC sets ξ,ζ,there exists a pair of disjoint HFO sets ω, ϱ such that ξ < ω, ζ < ϱ and Cl(ω) ∧ Cl(ϱ) = h0. Proof. (i) ⇒ (ii): Obvious. (ii) ⇒ (iii):Suppose that ξ < h1 − ζ for any HFMAO set h1 − ζ.By (ii) there exists HFO set ω such that ξ < ω < Cl(ω) < h1 − ζ.Clearly Cl(ω) ∧ ζ = h0 as Cl(ω) < h1 − ζ. By assuming ϱ = h1 − Cl(ω),we get ζ < ϱ < h1 − ω < h1 − ξ. Since h1 − ω is HFC set ζ < Cl(ϱ) < h1 − ω < h1 − ξ.Clearly,Cl(ϱ) ∧ ξ = h0 as Cl(ϱ) < h1 − ξ. It is evident that ω ∨ ϱ = h0. (iii) ⇒ (iv): By (iii) for any distinct HFO sets ω,ϱ such that ξ < ω,Cl(ω)∧ζ = h0 and ζ < ϱ,Cl(ϱ) ∧ ξ = h0. As Cl(ω) ∧ ζ = h0 , Cl(ϱ) ∧ ξ = h0 imply that Cl(ω) ∧ Cl(ϱ) = h0. A. Swaminathan, Cenap Ozel, Ibtesam Alshammari / Eur. J. Pure Appl. Math, 16 (3) (2023), 1381-1388 1386 (iv) ⇒ (i): Proof is easy and hence omitted. Theorem 2.13.. Every hesitant fuzzy Hausdorff m-compact space is hesitant fuzzy mini- mal hesitant fuzzy c-regular. Proof. Let X be a hesitant fuzzy Hausdorff m-compact. Suppose γ ∈ X is HFMIC set and xα ∈ X such that xα /∈ λ. Since X is hesitant fuzzy Hausdorff, for each xβ ∈ G, we have Gxβ , Hxβ disjoint HFO sets such that xα ∈ Gxβ ,xβ ∈ Hxβ . Let G = {Hxβ |xβ ∈ λ} ∨ {h1 − λ}. Then G is HFMAO cover of X by Lemma 2.3.. By hesitant fuzzy m- compactness of X, then we have a finite hesitant fuzzy s-refinement H of G . Let M = ∨{Λ ∈ H |Λ∧λ ̸= h0}. So M is an HFO set which contains λ. Let Λ1,Λ2...Λn be the only hesitant fuzzy members of H such that Λk ∧ λ ̸= h0,k ∈ Z+. For each k ∈ Z+,∃ xγ ∈ λ such that Λk < ̸ = Hxβγ ,k ∈ Z+. We put H = n ∧ k=1 Gxβγ . Then xα ∈ µ. It is easy to show that G ∧H = h0. Corollary 2.14.. A hesitant fuzzy Hausdorff m-compact space is hesitant fuzzy minimal c-normal. Proof. Let ξ,ζ be distinct HFMIC sets in hesitant fuzzy Hausdorff m-compact space X. By Theorem 2.13., X is hesitant fuzzy minimal c-regular. Hence for each xφ ∈ ξ, ∃ G,H HFO sets such that xφ ∈ G,ζ < H and G ∧H = h0. The collection G = {Gxφ |xφ ∈ ξ} ∨ {h1 − ξ} is a HFMAO cover of X by Lemma 2.3.. Now proceeding like the proof of Theorem 4.3, we get two HFO sets η and µ such that ξ < λ, ζ < µ and G ∧H = h0. Lemma 2.15.. If Y is a HFC(resp.HFO) subset of a HFTS X, then HFMIC (resp.HFMIO) sets in the subspace Y of X are HFMIC (resp.HFMIO) sets in X. Proof. Let ξ be a HFMIC set in Y , a HFC subset of a HFTS X. Evidently ξ is also HFC in X as ξ = η ∧ Y for any HFC set η in X . If possible, suppose we have a HFC set β in X such that β < ξ. Clearly β ∧ Y is HFC in Y such that β ∧ Y < β < ξ; either β ∧ Y = ξ or β ∧ Y = h0 as ξ is HFMIC in Y . β ∧ Y = ξ implies that β ∧ Y = β = ξ. Now it is enough to prove that β = h0 for β ∧ Y = h0. We see that β < ξ < Y as ξ is a hesitant fuzzy subset of Y . So we have β ∧ Y = β ̸= h0 if ξ ̸= h0. Hence β = h0. Similarly we can prove for HFO sets. Definition 2.18. A hesitant fuzzy subspace Y of a HFTS X is said to be hesitant fuzzy minimally closed(resp. hesitant fuzzy minimally open) invariant if HFMIC(resp.HFMIO) sets of Y are also HFMIC (resp.HFMIO) sets of X. Theorem 2.16.. Hesitant fuzzy minimally closed invariant subspaces of hesitant fuzzy minimal c-normal spaces are hesitant fuzzy minimal c-normal. Proof. Let ξ,ζ be two distinct HFMIC sets in Y , where Y is hesitant fuzzy minimally invariant subspace of a hesitant fuzzy minimal c-normal space X. Hence ξ,ζ are HFMIC REFERENCES 1387 sets in X. As X is hesitant fuzzy minimal c-normal space, ∃ η,µ distinct HFO sets in X such that ξ < η,ζ < µ and (Y ∧ η) ∧ (Y ∧ µ) = h0.That is Y ∧ η ; Y ∧ µ are distinct HFO sets in Y such that ξ < (Y ∧ η) and ζ < (Y ∧ µ). Corollary 2.17.. Each HFC subspace of a hesitant fuzzy minimal c-normal space is hes- itant fuzzy minimal c-normal. Proof. Using Lemma 2.15., we have to proceed like that of Theorem 2.16.. 3. Conclusion In recent times the notion of hesitant fuzzy minimal and maximal open sets have been important concepts in the literature. There are some family between hesitant fuzzy maximal and hesitant fuzzy minimal sets which is called as hesitant fuzzy mean open sets. When we are dealing with hesitant fuzzy compactness, we may have various covers to h1. In this paper, we have used particularly hesitant fuzzy maximal open cover for compactness. Further another new ideas namely hesitant fuzzy minimal c-regular and hesitant fuzzy minimal c-normal are extended with various properties. In future one can conclude and study numerous properties of connectedness and compactness in hesitant fuzzy topology. Therefore the hesitant fuzzy minimal, maximal and mean open sets play dominant role and in further study these notions can be investigated via various kinds of hesitant open sets. References [1] C. L. Chang, Fuzzy topological spaces, J.Math. Anal. Appl., 24(1968),182-190. [2] D. Deepak, B. Mathew, S.Mohn and H.A. Garg, Topological structure involving hesi- tant fuzzy sets, J. Intell.Fuzzy Syst.,2019,36,6401-6412. [3] D. Divakaran and S. J. John, Hesitant fuzzy rough sets through hesitant fuzzy rela- tions, Ann. Fuzzy Math. Inform.,2014,8,33-46. [4] J. Kim, Y. B. Jun, P. K. Lim, J. G . Lee and K. Hur, The category of hesitant H-fuzzy sets.,Ann.Fuzzy Math.Inform.,2019,18,57-74. [5] J. G. Lee and K. Hur, Hesitant fuzzy topological spaces, Mathematics,2020,8,188. [6] M. Sankari and C. Murugesan, Hesitant fuzzy cut-point spaces(Submitted). [7] A. Swaminathan and S. Sivaraja, Hesitant Fuzzy maximal and minimal clopen sets, Creative Mathematics and Informatics,Vol.31,No.2(2022). [8] A. Swaminathan and S. Sivaraja, Hesitant fuzzy paraopen and hesitant fuzzy mean open sets, J. Appl. and Pure Math., Vol. 4(2022), No.3-4, pp. 141-150. REFERENCES 1388 [9] A. Swaminathan and S.Sivaraja, Hesitant fuzzy minimal and maximal open sets, J. Appl. and Pure Math., Vol. 5(2023),1-2, 121-128.. [10] V. Torra, Hesitant fuzzy sets. Int. J. Intel. Sys., 2010,25,529-539. [11] L. A. Zadeh, Fuzzy sets, Information and control,8 (1965), 338-353.