The aim of this article is to introduce fuzzy minimal δ-open and fuzzy maximal δ-open sets in fuzzy topological space. Further, we investigate some properties with these new sets

Joseph S

Kirubaharan D. R.

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CREAT. MATH. INFORM.Volume 31 (2022), No. 2,Pages 183 - 188Online version at https://semnul.com/creative-mathematics/Print Edition: ISSN 1584 - 286X; Online Edition: ISSN 1843 - 441XDOI: https://doi.org/10.37193/CMI.2022.02.04Fuzzy Minimal and Maximal δ-Open SetsS. JOSEPH and D. R. KIRUBAHARANABSTRACT. The aim of this article is to introduce fuzzy minimal δ-open and fuzzy maximal δ-open sets infuzzy topological space. Further, we investigate some properties with these new sets.1. INTRODUCTIONZadeh [6] established fuzzy set in 1965 and Chang [2] introduced fuzzy topology in1968. Ittanagi and Wali [3] instigated the notions of fuzzy maximal and minimal opensets. The notion of fuzzy δ-open set introduced by Supriti Saha [5]. In this paper , weintroduce fuzzy minimal δ-open and fuzzy maximal δ-open sets. Further some of theirrelated results investigated.A proper nonempty fuzzy open set U of Xis said to be a fuzzy minimal open [3] set ifU and 0X are only fuzzy open sets contained in U .A proper nonempty fuzzy open set U of Xis said to be a fuzzy maximal open [3] set if1X and U are only fuzzy open sets containing U .A fuzzy subset K of a space X is called fuzzy regular open [1] (resp. fuzzy regularclosed) if K = Int(Cl(K)) (resp.K =Cl(Int(K))).The fuzzy δ-interior of a fuzzy subset K of X is the union of all fuzzy regular opensets contained in K. A fuzzy subset K is called fuzzy δ-open [5] if K = Intδ(K). Thecomplement of fuzzy δ-open set is called fuzzy δ-closed (i.e, K = Clδ(K)).2. FUZZY MINIMAL δ-OPEN SETSDefinition 2.1. A proper nonzero fuzzy δ-open set U in fts (X, τ) is said to be fuzzyminimal δ-open if fuzzy δ-open set contained in U is 0X or U .Lemma 2.1. Let (X, τ) be a fts.(i) If U1 is fuzzy minimal δ-open and U2 is fuzzy δ-open in X , then U1 ∩ U2 = 0X or U1 ⊂ U2.(ii) If U1 and U2 are fuzzy minimal δ-open , then U1 ∩ U2 = 0X or U1 = U2.Proof. (i) Let us assume that U2 is fuzzy δ-open in X such that U1 ∩ U2 ̸= 0X . Since U1 isfuzzy minimal δ-open , and U1 ∩ U2 ⊂ U1, then U1 ∩ U2 = U1 implies that U1 ⊂ U2.(ii) Suppose that U1 ∩ U2 ̸= 0X , then clearly from(ii), U1 ⊂ U2 and U2 ⊂ U1 as U1 andU2 are fuzzy minimal δ-open . Hence U1 = U2. □Theorem 2.1. Let U and Ui are fuzzy minimal δ-open sets for any i ∈ M . If U ⊆⋃i∈MUi, thenU = Uj for any j ∈ M .Received: 11.10.2021. In revised form: 24.01.2022. Accepted: 31.01.20222010 Mathematics Subject Classification. 54A40, 03E72.Key words and phrases. Fuzzy regular open set, Fuzzy δ-open set, Fuzzy minimal δ-open set, Fuzzy maximal δ-openset.Corresponding author: S. Joseph; jjalenjose@gmail.com183184 S. Joseph and D. R. KirubaharanProof. Suppose U ⊆⋃i∈MUi, then U = U ∩( ⋃i∈MUi)=⋃i∈M(U ∩ Ui). By deploying lemma2.1(ii), U ∩ Ui = 0X or U = Ui as U and Ui are fuzzy minimal δ-open sets. If U ∩ Ui = 0X ,then U = 0X which contradicts that U is a fuzzy minimal δ-open set. Hence if U ∩Ui ̸= 0Xthen U = Uj for any j ∈ M . □Theorem 2.2. If U and Ui are fuzzy minimal δ-open sets for any i ∈ M and U ̸= Ui, thenU ∩( ⋃i∈MUi)= 0X for any i ∈ M .Proof. Let U ∩( ⋃i∈MUi)̸= 0X , then U ∩ Ui ̸= 0X for any i ∈ M . By deploying lemma2.1(ii), U = Ui contradictory to U ̸= Ui. Hence U ∩( ⋃i∈MUi)= 0X . □Theorem 2.3. If Ui is a fuzzy minimal δ-open for any i ∈ M (|M | ≥ 2) and Ui ̸= Uj for anydistinct i, j ∈ M , then( ⋃i∈M\{j}Ui)∩ Uj = 0X for any j ∈ M .Proof. Let( ⋃i∈M\{j}Ui)∩ Uj ̸= 0X . Then⋃i∈M\{j}(Ui ∩ Uj) ̸= 0X ⇒ (Ui ∩ Uj) ̸= 0X . Bylemma 2.1(ii), Ui = Uj , a contradiction. Hence( ⋃i∈M\{j}Ui)∩Uj = 0X for any j ∈ M . □Theorem 2.4. If Ui is a fuzzy minimal δ-open for any i ∈ M, (|M | ≥ 2) and Ui ̸= Uj for anydistinct i, j ∈ M . If K is a proper fuzzy set of M , then( ⋃i∈M\KUi)⋂( ⋃s∈KUs)= 0X .Proof. Let( ⋃i∈M\KUi)⋂( ⋃s∈KUs)̸= 0X . It implies that⋃(Ui ∩ Us) ̸= 0X for i ∈ M\Kand s ∈ K implies that Ui∩Us ̸= 0X for some i ∈ M and s ∈ K. By lemma2.1(ii), Ui = Us,which is a contradiction. Hence( ⋃i∈M\KUi)⋂( ⋃s∈KUs)= 0X . □Theorem 2.5. If Ui is a fuzzy minimal δ-open for any i ∈ M such that Ui ̸= Uj for any distincti, j ∈ M . If S is a proper nonzero fuzzy set of M , then[ ⋃i∈M\kUi]∩[ ⋃k∈SUk]= 0X .Proof. Assume that ∪ [Ui ∩ Uk] ̸= 0X for i ∈ M\k,k ∈ S. Clearly, for some i ∈ M ,k ∈ S wehave [Ui ∩ Uk] ̸= 0X . By deploying lemma 2.1(ii) Ui = Uk, a contradiction. □Theorem 2.6. If Ui and Uk are fuzzy minimal δ-open sets for any i ∈ M and k ∈ S and if ∃ ann ∈ S such that Ui ̸= Un for any i ∈ M , then[ ⋃n∈KUn]̸⊂[ ⋃i∈MUi].Proof. Assume that ∃ an n ∈ S such that Ui ̸= Un for any i ∈ M , then[ ⋃n∈KUn]⊂[ ⋃i∈MUi].Fuzzy Minimal and Maximal δ-Open Sets 185⇒ Un ⊂[ ⋃i∈MUi]for some n ∈ K.⇒ Ui ̸= Un for any i ∈ M , by theorem 2.1, which is a contradiction. Hence[ ⋃n∈KUn]̸⊂[ ⋃i∈MUi]. □Theorem 2.7. If Ui is a fuzzy minimal δ-open for any i ∈ M such that Ui ̸= Uj for any distincti, j ∈ M , then[ ⋃k∈KUk]⫋[ ⋃i∈MUi]for any proper nonzero subset K of M .Proof. Let m ∈ M\K, then Um is a fuzzy minimal δ-open set of the family {Um|m ∈M\K} of fuzzy minimal δ-open sets. Clearly Um ∩[ ⋃k∈KUk]=⋃k∈K[Um ∩ Uk] = 0X . AlsoUm ∩[ ⋃i∈MUi]=⋃i∈M[Um ∩ Ui] = Um.If[ ⋃k∈KUk]=[ ⋃i∈MUi], then Um = 0X which is a contradiction that Um is a fuzzyminimal δ-open set. Hence[ ⋃k∈KUk]⫋[ ⋃i∈MUi]. □Theorem 2.8. If Ui is a fuzzy minimal δ-open set for any i ∈ M such that Ui ̸= Uj for anydistinct i, j ∈ M , then(i) Uj ⊂[ ⋃i∈M\{j}Ui]cfor some j ∈ M .(ii)⋃i∈M\{j}Ui ̸= 1X for any j ∈ M .Proof. (i) By hypothesis, Ui ̸= Uj for any distinct i, j ∈ M .By theorem 2.2,[ ⋃i∈MUi]∩ Uj = 0X which is true for any j ∈ M.⇒⋃i∈M[Ui ∩ Uj ] = 0X⇒ Ui ∩ Uj = 0X (By Lemma 2.1(ii))⇒ Ui ⊂ Ujc⇒⋃i∈M\{j}Ui ⊂ Ujc. Hence proved.(ii) Let j ∈ M such that⋃i∈M\{j}Ui = 1X⇒ Ui = 0X⇒ Ui is not a fuzzy minimal δ-open set, a contradiction. Hence⋃i∈M\{j}Ui ̸= 1X for anyj ∈ M . □Corollary 2.1. If Ui is a fuzzy minimal δ-open set for any i ∈ M such that Ui ̸= Uj for anydistinct i, j ∈ M , then Ui ∪ Uj ̸= 1X for any distinct i, j ∈ M .Proof. Similar to that of “Theorem 2.8 (ii).” □Theorem 2.9. If Ui is a fuzzy minimal δ-open sets for any i ∈ M such that Ui ̸= Uj for anydistinct i, j ∈ M , then Uj =[ ⋃i∈MUi]∩[ ⋃i∈M\{j}Ui]cfor any j ∈ M .186 S. Joseph and D. R. KirubaharanProof. For any j ∈ M ⇒[ ⋃i∈MUi]⋂[ ⋃i∈M\{j}Ui]c=[ ⋃i∈M\{j}Ui⋃Uj]∩[ ⋃i∈M\{j}Ui]c=[( ⋃i∈M\{j}Ui)∩( ⋃i∈M\{j}Ui)c]⋃[Uj ∩( ⋃i∈M\{j}Ui)c]= 0X ∪ Uj= Uj for any j ∈ M . □Proposition 2.1. Let G be a fuzzy minimal δ-open set. If xα ∈ G, then G ⊂ G1 for any fuzzyopen neighbourhood G1 of xα.Proof. Let G1 be an fuzzy δ-open neighborhood of xα such that G ̸⊂ G1. Clearly G∩G1 isan fuzzy δ-open such that G ∩ G1 ⊊ G and G ∩ G1 ̸= 0X . This implies that G is a fuzzyminimal δ-open set which a contradiction. □Proposition 2.2. Let G be a fuzzy minimal δ-open set. ThenG =⋂{G1 : G1 fuzzy e-open neighbourhood of xα for any xα ∈ G} .Proof. By deploying proposition 2.1 and as G is an fuzzy δ-open neighborhood of xα,we have G ⊂⋂{G1 : G1 fuzzy e-open neighbourhood of xα ⊂ G} . This completes theproof. □Theorem 2.10. Let G be a fuzzy minimal δ-open set. Then the following conditions are equivalent.(i) G is fuzzy minimal δ-open set.(ii)G ⊂ eCl(K) for any nonzero fuzzy subset K of G.(iii)eCl(G) = eCl(K) for any nonzero fuzzy subset K of G.Proof. (i) ⇒ (ii): By deploying “proposition 2.1”for any xα ∈ G and fuzzy δ-open neigh-borhood M of xα, we have K = (G∩K) ⊂ (M ∩K) for any proper nonzero fuzzy subsetK ⊂ G. Therefore, we have (M ∩K) ̸= 0X and xα ∈ eCl(K). It follows that G ⊂ eCl(K).(ii) ⇒ (iii): For any proper fuzzy subset K of G, eCl(G) ⊂ eCl(K). Also by(ii) eCl(G) ⊂eCl (eCl(K)) = eCl(K). Hence proved.(iii) ⇒ (i): Let us assume that G is not fuzzy minimal δ-open . Then there existsa proper fuzzy δ-open D such that D ⊂ G.Then ∃ yα ∈ G such that yα ̸∈ D. TheneCl({yα}) ∈ Dc implies that eCl({yα}) ̸= eCl(G), a contradiction. This completes ourproof. □3. FUZZY MINIMAL δ-OPEN SETS AND ITS PROPERTIESDefinition 3.2. A proper nonzero fuzzy δ-open set V of a fts (X, τ) is said to fuzzy maxi-mal δ-open if any fuzzy δ-open set which contains V is either V or 1X .Example 3.1. Let X = {a, b, c, d}. Then fuzzy sets F1 = 0.3a +0.4b +0.3c +0.4d ; F2 =0.5a +0.5b +0.5c +0.5d ; F3 =0.7a +0.6b +0.7c +0.6d ; are defined as follows: Consider thefuzzy topology τ = {0X ,F1,F2,F3, 1X}. Here F1 is fuzzy minimal δ-open and F3 fuzzymaximal δ-open set.Lemma 3.2. Let (X, τ) be a fts. Then(i) If V1 is a fuzzy maximal δ-open and V2 is fuzzy δ-open in X , then V1 ∪ V2 = 1X orV2 ⊂ V1.(ii) If V1 and V3 are fuzzy maximal δ-open sets, then either V1 ∪ V3 = 1X or V1 = V3.Proof. (i) Assume that V2 ̸⊂ V1. Clearly, V1 ⊂ (V1 ∪V2) a contrary to V1 is a fuzzy maximalδ-open set if V1 ∪ V2 ̸= 1X . Hence, V1 ∪ V2 = 1X .(ii) Let V1 and V3 are fuzzy maximal δ-open sets. Then from(i) V3 ⊂ V1 and V1 ⊂ V3implies that V1 = V3. □Fuzzy Minimal and Maximal δ-Open Sets 187Theorem 3.11. If V1, V2 and V3 are fuzzy maximal δ-open sets such that V1 ̸= V2 and(V1 ∩ V2) ⊂ V3, then either V1 = V3 or V2 = V3.Proof. Suppose that V1, V2 and V3 are fuzzy maximal δ-open sets with V1 ̸= V2 , (V1∩V2) ⊂V3 and if V1 ̸= V3, then(V2 ∩ V3) = V2 ∩ (V3 ∩ 1X)= V2 ∩ [V3 ∩ (V1 ∪ V2)], by lemma 3.2(ii)= V2 ∩ [(V3 ∩ V1) ∪ (V3 ∩ V2)]= [V2 ∩ V3 ∩ V1] ∪ [V2 ∩ V3 ∩ V2]= [V2 ∩ V1] ∪ [V2 ∩ V3]= V2 ∩ [V1 ∪ V3]= V2 ∩ 1X= V2(V2 ∩ V3) = V2 ⇒ V2 ⊂ V3. As V2 and V3 are fuzzy maximal δ-open sets, V3 ⊂ V2.Hence V2 = V3. □Theorem 3.12. For any distinct fuzzy maximal δ-open sets V1, V2, V3[V1 ∩ V2] ̸⊂ [V1 ∩ V3].Proof. Consider [V1 ∩V2] ⊂ [V1 ∩V3] for any distinct fuzzy maximal δ-open sets V1, V2 andV3. Then[V1 ∩ V2] ∪ [V2 ∩ V3] ⊂ [V1 ∩ V3] ∪ [V2 ∩ V3]= [V1 ∪ V3] ∩ V2 ⊂ [V1 ∪ V2] ∩ V3= 1X ∩ V2 ⊂ 1X ∩ V3= V2 is contained in V3a contradiction to V1, V2 and V3 are distinct . Hence [V1 ∩ V2] ̸⊂ [V1 ∩ V3]. □Remark 3.1. Proofs of “Theorem 3.13, Corollary 3.2, Theorem 3.14 and Theorem 3.15” aresimilar to proofs of “Theorem 2.8, Corollary 2.1, Theorem 2.9 and Theorem 2.7” respec-tively. Hence the proofs are omitted.Theorem 3.13. If Vi is a fuzzy maximal δ-open sets for any i ∈ M,M is a finite set and Vi ̸= Vjfor any distinct i, j ∈ M , then(i)[ ⋂i∈M\{j}Vi]c⊂ Vj for any j ∈ M(ii)⋂i∈M\{j}Vi ̸= 0X for any j ∈ M .Corollary 3.2. If Vi is a fuzzy maximal δ-open sets for any i ∈ M,M is a finite set and Vi ̸= Vjfor any distinct i, j ∈ M then,V i ∩ Vj ̸= 0X for any distinct i, j ∈ M .Theorem 3.14. If Vi is a fuzzy maximal δ-open sets for any i ∈ M,M is a finite set and Vi ̸= Vjfor any distinct i, j ∈ M , then Vj =[ ⋂i∈MVi]∪[ ⋂i∈M\{j}Vi]cfor any j ∈ M .Theorem 3.15. If Vi is a fuzzy maximal δ-open sets for any i ∈ M,M is a finite set and Vi ̸= Vjfor any distinct i, j ∈ M and if K is a proper nonzero fuzzy subset of M , then⋂i∈MVi ⫋⋂k∈KVk.Theorem 3.16. If Vi is a fuzzy maximal δ-open sets for any i ∈ M,M is a finite set and Vi ̸= Vjfor any distinct i, j ∈ M and if⋂i∈MVi is a fuzzy subset, then Vj is a fuzzy subset for any j ∈ M .188 S. Joseph and D. R. KirubaharanProof. By “Theorem 3.14”, we have Vj =[ ⋂i∈MVi]∪[ ⋂i∈M\{j}Vi]cfor any j ∈ M .Vj =[ ⋂i∈MVi]∪[ ⋃i∈M\{j}V ci].Since M is finite,⋃i∈M\{j}V ci is fuzzy δ-closed. Hence Vj is fuzzy δ-closed for any j ∈M . □Theorem 3.17. If Vi is a fuzzy maximal δ-open set for any i ∈ M,M is a finite set and Vi ̸= Vjfor any distinct i, j ∈ M . If⋂i∈MVi = 0X ,then {Vi/i ∈ M} is the set of all fuzzy maximal δ-opensets of fts X .Proof. Suppose that ∃ another fuzzy maximal δ-open Vk of a fts X such that Vk ̸= Vi,∀i ∈ M .Clearly, 0X =⋂i∈MVi =⋂i∈(M∪k)\{k}Vi ̸= 0X , by Theorem 3.13(ii), a contradiction.Hence {Vi/i ∈ M} is the family of all fuzzy maximal δ-open sets of fts X . □REFERENCES[1] Azad, K. K. On fuzzy semi-continuity, fuzzy almost continuity and fuzzy weakly continuity. J. Math. Anal.Appl. 82 (1981), 14–32.[2] Chang, C. L. Fuzzy topological spaces. J. Math.Anal.Appl. 24 (1968), 182–190.[3] Ittanagi, B. M.; Wali, R. S. On fuzzy minimal open and fuzzy maximal open sets in fuzzy topological spaces.Int. j. math. sci. appl. 1 (2011), no. 2[4] Joseph, S.; Kirubaharan, D. R.Fuzzy minimal and maximal δ-open sets(communicated).[5] Supriti, S. Fuzzy δ-continuous mappings. J. Math. Anal. Appl. 126 (1987), 130–142.[6] Zadeh, L. A. Fuzzy sets. Inf Comput., 8 (1965), 338–353.DEPARTMENT OF MATHEMATICSRESEARCH SCHOLARPRIST DEEMED TO BE UNIVERSITYTHANJAVUR,TAMIL NADU-613403, INDIAEmail address: jjalenjose@gmail.comDEPARTMENT OF MATHEMATICSPRIST DEEMED TO BE UNIVERSITYTHANJAVUR,TAMIL NADU-613403, INDIAEmail address: jagkans@yahoo.com

Fuzzy Minimal and Maximal δ-Open Sets

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Fuzzy Minimal and Maximal δ-Open Sets

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