Fuzzy Pre-semi-closed Sets

Transcription

1 BULLETIN of the Malaysian Mathematial Sienes Soiety Bull Malays Math Si So () 1() (008), Fuzzy Pre-semi-losed Sets 1 S Murugesan and P Thangavelu 1 Department of Mathematis, Sri S R Naidu Memorial College, Satur 66 0, India, Department of Mathematis, Aditnar College of Arts and Siene, Trihendur India 1 satturmuruges@gmailom; pthangavelu004@yahoooin Abstrat In this paper, we introdue fuzzy pre-semi-losed sets in fuzzy topologial spaes and investigate their properties Using fuzzy pre-semi-losed sets, equivalenes of fuzzy regular open sets are established As appliations to fuzzy pre-semi-losed sets, we introdue fuzzy spaes with new kinds of separation axioms, namely, fuzzy pre-semi-t 1 spaes, fuzzy pre-semi-t 4 spaes and fuzzy semi-pre-t 1 spaes and haraterize them 000 Mathematis Subjet Classifiation: 54A40 Key words and phrases: Fuzzy generalized losed sets, fuzzy semi-pre-losed sets, fuzzy T 1 spae, fuzzy semi-t 1 spae 1 Introdution and Preliminaries Reently, Veera Kumar [1] introdued pre-semi-losed sets in risp topologial spaes Extending this onept to fuzzy topologial spaes(fts), we define a new lass of fuzzy generalized sets namely, fuzzy pre-semi-losed sets and investigate their properties In this setion, we list out the definitions and the results whih are needed in sequel Throughout this paper, f ts X denote a fuzzy topologial spae (X, τ), in the sense of Chang [6] Fuzzy sets in X will be denoted by λ, µ, ν, η For a fuzzy set λ, the operators fuzzy losure and fuzzy interior are denoted and defined by Clλ = µ : µ λ, 1 µ τ} andintλ = µ : µ λ, µ τ} if X is a finite set and X = a i : i = 1,,,k}, where k is a positive integer Fuzzy sets in X may represented as λ = α 1 + α + + α k a 1 a a k and 0 α 1, α,,α k 1 We do not assume any meaning to symbols used unless defined here The following lemmas are well-known Lemma 11 [, 4, 1] In a fuzzy topologial spae (X, τ), (i) every fuzzy regular open set is fuzzy open Reeived: June 16, 007; Revised: June 17, 008

2 4 S Murugesan and P Thangavelu (ii) every fuzzy open set is fuzzy α-open (iii) every fuzzy α-open set is both fuzzy semiopen and fuzzy preopen (iv) every fuzzy semiopen set is fuzzy semi-preopen (v) every fuzzy preopen set is fuzzy semi-preopen Lemma 1 A fuzzy set λ in a fuzzy topologial spae (X, τ) is (i) fuzzy regular losed [] (simply Fr-losed) if λ = Cl Intλ (ii) fuzzy α-losed [4] (simply Fα-losed) if Cl Int Cl λ λ (iii) fuzzy pre-losed [4] (simply Fp-losed) if Cl Int λ λ (iv) fuzzy semi-losed [] (simply Fs-losed) if Int Cl λ λ (v) fuzzy semi-pre-losed [1] (simply Fsp-losed) if Int Cl Int λ λ Notation 11 Let (X, τ) be a fuzzy topologial spae Then the family of fuzzy regular losed (resp fuzzy semi-losed, fuzzy pre-losed, fuzzy semi-pre-losed) sets in X, may be denoted by its adjetive as r(resp α, s, p, sp) Definition 11 [, 4, 1] Let λ be a fuzzy set in a fuzzy topologial spae (X, τ) Then fuzzy β-losure of λ is denoted and defined by β Clλ = µ : λ µ, µ β} where β r, α, s, p, sp} The following three lemmas are well-known Lemma 1 Let λ be a fuzzy set in a fuzzy topologial spae (X, τ) Then (i) sp Clλ s Cl λ α Cl λ Cl λ r Cl λ, (ii) sp Clλ p Cl λ α Cl λ Lemma 14 Let λ be a fuzzy set in a fuzzy topologial spae (X, τ) Then (i) α Cl λ = λ Cl IntCl λ (ii) s Clλ = λ IntCl λ (iii) p Clλ λ Cl Intλ (iv) sp Clλ λ IntCl Intλ Lemma 15 Let λ be a fuzzy set in a fuzzy topologial spae (X, τ) Then (i) 1 sp Int(µ) = sp Cl(1 µ) and 1 sp Clµ = sp Int(1 µ) (ii) sp Cl(sp Cl µ) = sp Clµ Definition 1 A fuzzy set λ in a fuzzy topologial spae (X, τ) is alled: (i) [] Fuzzy generalized losed set if Cl λ µ whenever λ µ and µ is fuzzy open We briefly denote it as Fg-losed set (ii) [8] Fuzzy semi-generalized losed set if s Cl λ µ whenever λ µ and µ is fuzzy semiopen We briefly denote it as Fsg-losed set (iii) [9] Fuzzy generalized strongly losed set if α Cl λ µ whenever λ µ and µ is fuzzy open We briefly denote it as Fgα-losed set (iv) [9] Fuzzy generalized almost strongly semi-losed set if α Cl λ µ whenever λ µ and µ is fuzzy α-open We briefly denote it as Fαg-losed set (v) [11] Fuzzy semi-pre-generalized losed set if sp Cl λ µ whenever λ µ and µ is fuzzy semi-pre-open We briefly denote it as Fspg-losed set (vi) [10] Regular generalized fuzzy losed set if Clλ µ whenever λ µ and µ is fuzzy regular open We briefly denote it as rfg-losed set

3 Fuzzy Pre-semi-losed Sets 5 (vii) [5] Fuzzy pre-generalized losed set if p Cl λ µ whenever λ µ and µ is fuzzy pre-open We briefly denote it as F pg-losed set Similarly, we may define: (viii) Fuzzy generalized semi-losed set if s Clλ µ whenever λ µ and µ is fuzzy open We briefly denote it as Fgs-losed set (ix) Fuzzy generalized pre-losed set if p Clλ µ whenever λ µ and µ is fuzzy open We briefly denote it as Fgp-losed set (x) Fuzzy generalized semi-pre-losed set if sp Clλ µ whenever λ µ and µ is fuzzy open We briefly denote it as Fgsp-losed set Definition 1 A fuzzy set λ in a fuzzy topologial spae (X, τ) is alled fuzzy generalized open set (briefly, Fg-open) if 1 λ is fuzzy generalized losed Similarly, fuzzy generalized semi-open, fuzzy semi-generalized open, fuzzy generalized strongly open, fuzzy generalized almost strongly semi-open, fuzzy generalized semi-pre-open, fuzzy semi-pre-generalized open, fuzzy generalized pre-open, fuzzy pre-generalized open sets are defined Proposition 11 In a fuzzy topologial spae (X, τ), the following hold and the onverse of eah statement is not true: (i) [] Every fuzzy losed set is Fg-losed (ii) [8] Every fuzzy semi-losed set is Fsg-losed (iii) [9] Every fuzzy α-losed set is Fαg-losed (iv) [5] Every fuzzy pre-losed set is Fpg-losed (v) [11] Every fuzzy semi-pre-losed set is Fspg-losed Definition 14 A fuzzy topologial spae (X, τ) is said to be a (i) [] fuzzy-t 1 (ii) [8] fuzzy semi-t 1 (iii) [5] fuzzy pre-t 1 Fuzzy Pre-semi-losed Sets spae if every Fg-losed set is fuzzy losed spae if every Fsg-losed set is fuzzy semi-losed spae if every Fpg-losed set is fuzzy pre-losed In this setion, we define a new lass of fuzzy generalized losed sets alled a fuzzy pre-semi-losed sets and study its properties Definition 1 Let µ be a fuzzy set in a fts (X, τ) Then λ is alled a fuzzy pre-semi-losed set X if sp Cl µ λ, whenever µ λ and λ is a Fg-open set in X The following proposition asserts that the lass of fuzzy pre-semi-losed set ontains the lass of fuzzy semi-pre-losed sets Proposition 1 Every fuzzy semi-pre-losed set in a fts (X, τ) is fuzzy pre-semilosed Proof Let µ be a fuzzy semi-pre-losed set in a fts (X, τ) Suppose that µ λ and λ is a Fg-open set in X Sine sp Cl µ = µ, it follows that sp Clµ = µ λ, and hene µ is fuzzy pre-semi-losed in X The reverse impliation in the above proposition is not true as seen in the following example

4 6 S Murugesan and P Thangavelu Example 1 Consider the f ts (X, τ), where X = a, b, } and τ = 0, 1, µ = 07 a + 0 b + 1, λ = 07 a + 0 b + 0 } Fuzzy losed sets in X are: 0, 1, µ = 0 a + 07 b + 0, λ = 0 a + 1 b + 1 So the family of Fg-losed sets is 0, 1, µ, λ, α 1 either α 1 > 07 or α > 0 Hene the family of Fg-open sets is 0, 1, µ, λ, α 1 either α 1 < 0 or α < 07 Now ν = 1 a + 0 b + 0 is not a fuzzy semi-pre-losed set in X, for Int ν = λ and so, Int Cl Int ν = Int Cl λ = 1 > ν Moreover, ν is fuzzy pre-semi-losed Indeed, let ν η and η be F g-open in X Then η = 1 and sp Cl ν η From the sueeding two examples, it an be seen that fuzzy pre-semi-losedness is independent from F g-losedness, F gs-losedness and F gα-losedness Example Consider the f ts (X, τ), where X = a, b, } and τ = 0, 1, λ = 09 a + 0 b + 0 } Fuzzy losed sets are: So the family of Fg-losed sets in X is 0, 1, λ = 01 a + 08 b + 1 0, 1, λ, and α 1 b + α either 09 < α 1 or 0 < α } Hene the family of Fg-open sets is 0, 1, λ, α 1 either α 1 < 01 or α < 08 Now ν = 09 a + 01 b + 0 is a fuzzy pre-semi-losed set in X, for if ν η and η is Fg-open set in X, then η = 1 and hene sp Cl ν η But, Int Clν = 1, ν λ and Int Cl ν = 1 > λ, so ν is not a Fgs-losed set in X and hene by Lemma 1, it is neither Fg-losed nor F gα-losed

5 Fuzzy Pre-semi-losed Sets 7 Example Consider the fts (X, τ) : X = a, b, } and τ = 0, 1, λ = 1 a + 0 b + 0 } Fuzzy losed sets are: 0, 1, λ = 0 a + 1 b + 1 So the family of Fg-losed sets in X is 0, 1, α 1 either α 1 > 0 or α > 0 Hene the family of Fg-open sets in X is 0, 1, λ, α 1 either α 1 < 1 or α < 1 Now ν = 1 a + 1 b + 0 is not a fuzzy pre-semi-losed set in X; for if ν ν and ν is Fg-open set, but, Int Cl Int ν = 1, and hene sp Clν = 1 > ν But, ν is Fg-losed set in X and hene it is Fgα-losed and Fgs-losed Proposition Every fuzzy pre-semi-losed set in a f ts (X, τ) is F gsp-losed Proof Let ν be a fuzzy pre-semi-losed set in a fts (X, τ) Suppose that ν λ and λ is a fuzzy open set in X Then, sp Clν λ and λ is Fg-open in X and hene ν is F gsp-losed in X Remark 1 In Example, fuzzy set ν is also a Fgsp-losed set but not a fuzzy pre-semi-losed set Thus, the onverse of the above proposition is not true Proposition Union of two fuzzy pre-semi-losed sets in a fts need not be fuzzy pre-semi-losed Proof Consider the fts (X, τ) : X = a, b, } and τ = 0, 1, λ = 1 a + 05 b + 0 } Fuzzy losed sets are: 0, 1, λ = 0 a + 05 b + 1 So the family of Fg-losed sets in X is 0, 1, λ, α 1 b + α Hene the family of Fg-open sets in X is Let } where α 0 0, 1, λ, α 1 b + α where 1 α µ = 1 a + 0 b + 0 and ν = 0 a + 05 b + 0 be fuzzy sets in X Now, sine Int Cl Int µ = 0 µ and Int Cl Int ν = 0 ν, µ and ν are fuzzy semi-pre-losed sets in X and hene by Proposition 1, µ and ν }

6 8 S Murugesan and P Thangavelu are fuzzy pre-semi-losed sets in X But µ ν is not fuzzy pre-semi-losed set in X Indeed, µ ν = λ and λ is Fg-open, but sp Cl λ = 1 > λ Proposition 4 Let µ be a fuzzy set in a fts (X, τ) If µ is Fg-open and fuzzy pre-semi-losed, then µ is fuzzy semi-losed Proof Sine µ µ is Fg-open, it follows that µ Int Cl Int µ sp Cl µ µ Hene, Int Cl Int µ µ and µ is fuzzy semi-losed As a onsequene of Lemma 11, Lemma 1, Proposition and Proposition 1, we get Figure 1 Fα losed F losed Fαg losed Fg losed Fp losed Fgα losed Fs losed Fpg losed Fsg losed Fgp losed Fsp losed Fgs losed Fspg losed Fgsp losed Fps losed Figure 1 The following proposition haraterizes fuzzy regular open sets Proposition 5 Let µ be a fuzzy set in a fts (X, τ) Then the following are equivalent:

7 Fuzzy Pre-semi-losed Sets 9 (i) µ is fuzzy regular open (ii) µ is fuzzy open and fuzzy pre-semi-losed (iii) µ is fuzzy open and Fgsp-losed Proof (i) (ii): Let µ be fuzzy regular open in a fts (X, τ) Then it is both fuzzy open and fuzzy semi-losed and hene by Lemma 11 and Proposition 1, it is fuzzy pre-semi-losed (ii) (iii): Let µ be fuzzy open and fuzzy pre-semi-losed Then, by Proposition, it is Fgsp-losed (iii) (i): Let µ be fuzzy open and Fgsp-losed Then, µ µ and µ is fuzzy open and so, µ Int Cl Int µ sp Clµ µ Hene, Int Cl Int µ µ Sine µ is fuzzy open it follows that Int Clµ µ = Int µ Int Cl µ Hene µ is fuzzy regular open Proposition 6 Let µ be a fuzzy pre-semi-losed set in a fts (X, τ) If µ is a fuzzy set in X suh that µ λ sp Clµ, then λ is also fuzzy pre-semi-losed Proof Let λ η and η be Fg-open in X Then µ η and sine µ is fuzzy pre-semilosed, it follows from Lemma 15(ii) that sp Clλ sp Cl(sp Clµ) = sp Cl µ η Hene, λ is fuzzy pre-semi-losed Definition A fuzzy set µ in a fts (X, τ) is said to be fuzzy pre-semi open if 1 µ is fuzzy pre-semi-losed Proposition 7 Let µ be a fuzzy pre-semi-losed set in a fts (X, τ) If λ is a fuzzy set in X suh that sp Int µ λ µ, then λ is also a fuzzy pre-semi open Proof Now, 1 µ is a fuzzy pre-semi-losed set in X and λ is a fuzzy set in X suh that 1 µ 1 λ 1 sp Int µ By Lemma 1, 1 µ 1 sp Int µ = sp Cl(1 µ) Hene, by Proposition 6, 1 λ is also fuzzy pre-semi-losed and λ is fuzzy pre-semi open Fuzzy Pre-semi-T 1 Spaes As appliations of fuzzy pre-semi-losed sets, three fuzzy spaes namely, fuzzy pre-semi-t 1 spaes, fuzzy pre-semi-t spaes and fuzzy semi-pre-t 1 spae are introdued 4 Definition 1 A fuzzy topologial spae (X, τ) is alled a (i) fuzzy pre-semi-t 1 spae if every fuzzy pre-semi-losed set in it is fuzzy semipre-losed, (ii) fuzzy semi-pre-t 1 spae if every Fgsp-losed set in it is fuzzy semi-prelosed Proposition 1 Every fuzzy semi-pre-t 1 spae is a fuzzy pre-semi-t 1 spae Proof Let (X, τ) be a fuzzy semi-pre-t 1 spae and λ be a fuzzy pre-semi-losed set in X By Proposition, λ is Fgsp-losed set in X and hene it is fuzzy semi-prelosed set in X Thus, (X, τ) is fuzzy pre-semi-t 1 spae The onverse of the above proposition is not valid as seen in the following example

8 0 S Murugesan and P Thangavelu Example 1 Consider the fts (X, τ) : X = a, b, } and τ = 0, 1, λ = 1 a + 0 b + 0 } Fuzzy losed sets are: 0, 1, λ = 0 a + 1 b + 1 So the family of Fg-losed sets in X is 0, 1, α 1, either α 1 0 or α 0 Hene the family of Fg-open sets in X is 0, 1, λ, α 1, either α 1 or α 1 Now ν = 1 a + 0 b + 1 is not a fuzzy pre-semi-losed sets in X Indeed, ν η and η is Fg-open implies that η = 1 and hene, sp Cl ν η But, ν is not fuzzy semi-pre-losed; for sp Clν = 1 > ν Thus, X is not a fuzzy semi-pre-t 1 spae However, X is a fuzzy pre-semi-t 1 spae Indeed, every fuzzy pre-semi-losed set in X is fuzzy semi-pre-losed For: Case 1 Suppose that µ is F g-open Then µ is fuzzy pre-semi-losed implies that sp Cl µ µ and hene sp Cl µ = µ So, µ is fuzzy semi-pre-losed Case Suppose that µ is not Fg-open Then µ = α 1 a + 1 b + 1 where 0 α 1 < 1, and 1 is the only Fg-open set ontaining µ and hene µ is fuzzy pre-semi-losed Moreover, µ is fuzzy semi-pre-losed Indeed, Int Cl Int µ = 0 < µ Definition Fuzzy topologial spae (X, τ) is alled a fuzzy semi-pre-t 1 spae if every Fgsp-losed set in it is fuzzy pre-semi-losed Proposition Every fuzzy semi-pre-t 1 spae is a fuzzy semi-pre-t 1 spae Proof Let (X, τ) be a fuzzy semi-pre-t 1 spae and µ be a Fgsp-losed set in (X, τ) Then µ is fuzzy semi-pre-losed and hene by Proposition 1, is fuzzy pre-semilosed Thus, (X, τ) is a fuzzy semi-pre-t 1 spae It will be an interesting exerise to prove that the onverse of the above proposition is not valid Definition A fuzzy topologial spae (X, τ) is alled a fuzzy pre-semi-t 4 spae if every fuzzy pre-semi-losed set in it is fuzzy pre-losed Proposition Every fuzzy pre-semi-t 4 spae is a fuzzy pre-semi-t 1 spae Proof Let (X, τ) be a fuzzy pre-semi-t spae and λ be a fuzzy pre-semi-losed set 4 in (X, τ) Then, λ is fuzzy pre-losed Thus, (X, τ) is a fuzzy pre-semi-t 1 spae Converse of the above proposition is not true as seen in the following example

9 Fuzzy Pre-semi-losed Sets 1 Example Let (X, τ) be a fts, where X = a, b, } and τ = 0, 1, λ 1 = 1 a + 0 b + 0, λ = 0 a + 1 b + 0, λ = 1 a + 1 b + 0 } Fuzzy losed sets in (X, τ) are 0, 1, λ 1 = 0 a + 1 b + 1, λ = 1 a + 0 b + 1, λ = 0 a + 0 b + 1 If µ is Fg-losed, then µ λ implies Cl µ λ whenever λ is fuzzy open Thus, F g-losed sets in (X, τ) are: 0, 1, λ 1 = 0 a + 1 b + 1, λ = 1 a + 0 b + 1, λ = 0 a + 0 b + 1 where α 0 So, the family of Fg-open sets in (X, τ) is 0, 1, α 1 where α 1 and α 1 b + α It is enough to prove that, if µ is not fuzzy semi-pre-losed then it is not fuzzy pre-semi-losed and there is a fuzzy pre-semi-losed set whih is not pre-losed Let µ 0 be a fuzzy set in X Then, Intµ = 0 λ 1 λ λ 1, Cl Int µ = 0 λ λ 1 1 and IntCl Intµ = So, µ is not fuzzy semi-pre-losed if λ µ In that ase, µ is also not fuzzy presemi-losed For µ is Fg-open and µ µ But sp Cl µ µ Int Cl Int µ = 1 > µ Thus X is a pre-semi-t 1 spae But it is not a fuzzy pre-semi-t spae Indeed, 4 µ = 1 a + 0 b + 0 is a fuzzy semi-pre-losed and hene it is fuzzy pre-semi-losed but it is not fuzzy pre-losed, as Cl Int µ µ 0 λ 1 λ Conlusion In this paper, by the introdution of fuzzy pre-semi-losed sets, we have equivalenes of fuzzy regular open sets and fuzzy spaes with new separation axioms, namely, fuzzy semi-pre-t 1 spae, pre-semi-t spae and fuzzy pre-semi-t 1 spae 4 In risp topology onstrution of ounter examples is easy, but here that will be a rewarding exerises Aknowledgement We aknowledge the suggestions of the referees for the improvement of the paper

10 S Murugesan and P Thangavelu Referenes [1] K M Abd El-Hakiem, Generalized semi-ontinuous mappings in fuzzy topologial spaes, J Fuzzy Math 7()(1999), [] K K Azad, On fuzzy ontinuity, fuzzy almost ontinuity and fuzzy weakly ontinuity, J Math Anal Appl 8(1981), 14 [] G Balasubramanian and P Sundaram, On some generalizations of fuzzy ontinuous funtions, Fuzzy Sets and Systems 86(1)(1997), [4] A S Bin Shahna, On fuzzy strong semi-ontinuity and fuzzy pre-ontinuity, Fuzzy Sets and Systems 44(1991), 14 [5] M Caldas, G Navagi and R Saraf, On some funtions onerning fuzzy pg-losed sets, Proyeiones 5()(006), 6 71 [6] C L Chang, Fuzzy topologial spaes, J Math Anal Appl 4(1968), [7] George J Klir and Bo Yuan, Fuzzy Sets and Fuzzy Logi Theory and Appliations, Seventh Printing, Prentie-Hall, In, 1995 [8] H Maki et al, Generalized losed sets in fuzzy topologial spae, I, Meetings on Topologial Spaes Theory and its Appliations (1998), 6 [9] O Bedre Ozbakir, On generalized fuzzy strongly semilosed sets in fuzzy topologial spaes, Int J Math Math Si 0(11)(00), [10] J H Park and J K Park, On regular generalized fuzzy losed sets and generalizations of fuzzy ontinuous funtions, Indian J Pure Appl Math 4(7)(00), [11] R K Saraf, G Navalagi and Meena Khanna, On fuzzy semi-pre-generalized losed sets, Bull Mal Mat Si So 8(1)(005), 19 0 [1] S S Thakur and S Singh, On fuzzy semi-preopen sets and fuzzy semi-pre ontinuity, Fuzzy Sets and Systems 98(1998), 8 91 [1] M K R S Veera Kumar, Pre-semi-losed sets, Indian J Math 44() (00),