The Joural of Mathematics ad Computer Sciece Available olie at http://www.tjmcs.com The Joural of Mathematics ad Computer Sciece Vol.2 No.3 2) 475-482 ON THE FUZZY METRIC SPACES Received: July 2, Revised: October 2 Olie Publicatio: Jauary 2 G. A. Afrouzi, S. Shakeri 2, S. H. Rasouli 3 afrouzi@umz.ac.ir s.shakeri@iauamol.ac.ir s.h.rasouli@it.ac.ir Abstract I this paper we defie complete fuzzy metric space ad proved that a fuzzy topologically complete subset of a fuzzy metric space is a G set ad prove that a coverse of Sierpisky theorem by showig that ay G set i a complete metric space is a topologically complete fuzzy metrizable space Alexadroff Theorem). Keywords: fuzzy metric spaces, complete fuzzy metrizable space, Alexadroff theorem. Itroductio The theory of fuzzy sets was itroduced by L.Zadeh i 965 [2]. After the pioeerig work of Zadeh, there has bee a great effort to obtai fuzzy aalogues of classical theories. Amog other fields, a progressive developmets is made i the field of fuzzy topology. Oe Departmet of Mathematics, Faculty of Basic Scieces, Mazadara Uiversit Babolsa Ira 2 Departmet of Mathematics, Islamic Azad Uiversity Aytollah Amoli Brach, Amol, Ira 3 Departmet of Mathematics, Faculty of Basic sciece, Babol Noshirvai Uiversity of Techolog Babol, Ira. 475
of the most importat problems i fuzzy topology is to obtai a appropriate cocept of fuzzy metric space. This problem has bee ivestigated by may authors from differet poits of view. I particula George ad Veeramai [2] have itroduced ad studied a otio of fuzzy metric space. Furthermore, the class of topological spaces that are fuzzy metrizable agrees with the class of metrizable- topological spaces see[2] ad [4]). This result permits Gregori ad Romaguera to restate some classical theorems o metric completeess ad metric pre) compactess i the realm of fuzzy metric spaces [4], [5] ad [6]. I this paper we cosider modified fuzzy metric space itroduced by George ad Veeramai [2], [3]. I [], we proved that a fuzzy topologically complete subset of a fuzzy metric space is a G set Sierpisky Theorem). I this paper we prove a coverse of Sierpisky theorem by showig that ay G set i a complete metric space is a topologically complete fuzzy metrizable space Alexadroff Theorem). Defiitio.. A biary operatio * : [, ] [, ] [,] is a cotiuous t-orm if it satisfies the followig coditios. ) * is associative ad commutative, 2) * is cotiuous, 3) a * = a for all a [,], 4) a * b c*d wheever a c ad b d, for each a, b, c, d [,]. Two typical examples of cotiuous t-orm are a*b =ab ad a*b=mi a,b). The cocept of fuzzy metric space is defied by George ad Veeramai as follows [2]. Defiitio.2. A 3-tuple X, M, *) is called a fuzzy metric space if X is a arbitary o empty) set, * is a cotiuous t-orm, ad M is a fuzzy set o 2 X, ), satisfyig the followig coditios for each z X ad t,s >, ) M >, 2) M = if ad oly if x= 3) M = M, 4) * M z, M z,t+, 5) M.):, ) [,] is cotiuous. Let X, M, *) be a fuzzy metric space. For t>, the ope ball B with ceter radius x X ad 476
<r < is defied by B = { y X : M r}. Let X, M,*) be a fuzzy metric space. Let be the set of all A X with x A if ad oly if there exist t> ad <r< such that B A. The is a topology o X iduced by the fuzzy metric M). This topology is Hausdorff ad first coutable. A sequece {x } i X coverges to x if ad oly if Mx, as, for each t>. It is called a Cauchy sequece if for each ad t>, there exits such that M x, x m, >- for each,m. The fuzzy metric space X,M,*) is said to be complete if every Cauchy sequece is coverget. A subset A of X is said to be F-bouded if there exists t> ad <r< such that M> r for all ya. Theorem.3. I a fuzzy metric space every compact set is closed ad F-bouded. Theorem.4. I a fuzzy metric space every compact set is complete. Corollary.5. Every closed subset of a complete fuzzy metric space is complete. 2. Mai Result Lemma 2.. Let X,M,*) be a fuzzy metric space ad let [,) the there exists a fuzzy metric m o X such that m, for each y X ad t> ad m ad M iduce the same topology o X. Proof. We defie m=max {, M }. We claim that m is fuzzy metric o X. The properties of ), 2), 3) ad 5) are immediate from the defiitio. For triagle iequalit suppose that z X ad t,s>. The mz,t+ ad so mz,t+ m * my.z. whe either m= or mz,=. The oly remaiig case is whe m= M> ad mz, =Mz,>. But Mz,t+ M * Mz, ad mz,t+ Mz,t+ ad so mz,t + Mz,t+ ad so mz,t+ m * mz,. Thus m is a fuzzy metric o X. It oly remais to show that the topology iduced by m is the same as that iduced by M. But we have mx, if ad oly if {, M x, } if ad oly if Mx,, for each t>, ad we are doe. The fuzzy metric m i above lemma is said to be bouded by. Defiitio 2.2. Let X,M,*) be a fuzzy metric space, xx ad A X. 477
We defie DA, = sup {M:yA}t>). Note that DA, is a degree of closeess of x to A at t. Defiitio 2.3. A topological space is called a topologically complete) fuzzy metrizable space if there exists a topologically complete) fuzzy metric iducig the give topology o it. t Example. Let X=,]. The fuzzy metric space X,M,.) where M= stadard fuzzy t x y metric, see[2]) is ot complete because the Cauchy sequece {/} i this space is ot coverget. t Now, if we cosider triple X,m,.) where m. It is straightforward to t x y x y show that X,m,.) is a fuzzy metric space, ad that is complete. Sice, x ted to x with respect to fuzzy metric if ad oly if x x if ad oly if x ted to x with respect to fuzzy metric m, the M ad m are equivalet fuzzy metrics. Hece the fuzzy metric space X, M,.) is topologically complete fuzzy metrizable. Lemma 2.4. Fuzzy metrizability is preserved uder coutable Cartesia product. Proof. Without loss of geerality we may assume that the idex set is N Let { X, m,*) : N} be a collectio of fuzzy metrizable spaces. Let be the topology iduced by m o X for N ad let X, ) be the Cartesia product of { X, ) : N} with product topology. We have to prove that there is a fuzzy metric m o X which iduces the topology. By the ) ) above lemma, we may suppose that m is bouded by, i.e. ) m x,y,= max{-, M x, y, }. Poits of X N X are deoted as sequeces x x } with x X for N. Defie m= { m t>, a a * a *... a ). 2 m m x,y,, for each y X ad i ) First ote that m is well defied sice a ) is decreasig ad bouded the coverges i to, ). Also m is a fuzzy metric o X because each m is a fuzzy metric. Let u be the topology iduced by fuzzy metric m.we claim that u coicides with. If G u ad x = {x } G, the there exists < r< ad t> such that B G. For each <r<, we ca fid a sequece { } i,) ad a positive iteger N such that. N ) * N ) ) r. 478
For each =, 2,, N, let V B x,,, where the ball is with respect to fuzzy metric m. Let V =X for > N. Put V V N, the xv ad V is a ope set i the product topology o X. Furthermore V B, sice for each y V m m x, y, N m x, y, * m x N, y, N )* N ) ) r. Hece V B G. Therefore G is ope i the product topology. Coversely suppose G is ope i the product topology ad let x { x} G. choose a stadard basic ope set V such that xv ad V G. Let V V N, where each V is ope i X ad V =X for all >N. For =, 2,, N, let r = D x, X - V,, if r X V, ad ),otherwise. Let r r, r 2,..., r N }. mi{ B V. If y { y} B, the m m x so, y, >- r ad We claim that m x, y, r r for each =, 2,, N. The y, for =,2,, N. Also for >N, V y V X. Hece y V ad so B V G. Therefore G is ope with respect to the fuzzy metric topology ad r u. Hece r ad u coicide. Theorem 2.5. metrizable space. A ope subspace of a complete fuzzy metrizable space is a complete fuzzy Proof. Let X, M, *) be a complete fuzzy metric space ad G a ope subspace of X. If the restictio of M to G is ot complere we ca replace M o G by othere metric as follows. Defie f ) : G, R by f = D X G, f is udefied if X- G is empt but the there is othig to prove.) Fix a arbitary s> ad for y G defie. m M * M f, f,, 479
for each t>. We claim that m is fuzzy metric o G. The properties ), 2), 3) ad 5) are immediate from the defiitio. For triagle iequalit suppose that x, z G ad t, s, u >, the. m * m y,z, u) = = M* Mf, f,)* Mz,u)* Mf, fz,,u)) = M* Mz,u))* Mf, f,* Mf, fz,,u)) Mz,t+u) * M f, fz,, t+u)=mz,t+u). We show that m ad M are equivalet fuzzy metrics o G. We do this by showig that mx, if ad oly if Mx,. Sice m M for all y G ad t>, Mx, whewver mx,. To prove the covers, let Mx,, we kow from [7] Propositio M is cotiuous fuctio o X X, ), the sice. lim D X G, lim = M sup{ M x, : y G}) limm x, Therefore lim Dx, X-G, > D X-G,. O the other had, there exists a such that for every we have. D X G, * ) M y,. The lim Dx, X-G, M y, Sup { M : y X G} DX-G,. Therefore lim x This implies, X G, D X G,. Mfx,, f,. y X G ad Hece m x,. Therefore m ad M are equivalet. Next we show that m is a complete fuzzy metric. Suppose that {x } is a Cauchy sequece i G with respect to m. Sice for each m, ad t> m xm, M xm,, therefore {x } is also a Cauchy sequece with respect to M. By completeess of X, M,*), {x } coverges to poit p i X. We claim that, if p X G ad M x, p, D x, X G,, the p G. Assume otherwise, the for each M x, p, D x, X G,, 48
Therefore D x, X G, M x That is f x, M x, p,, p, for each t>. Therefore as.o the other had, M f, f xm,, m xm,,for every, for every t> we get f x,. I particular, f x, m,, that is { f x, } is a F-bouded sequece. This cotradictio shows that p G. Hece {x } coverges to p with respect to m ad G,m,*) is a complete fuzzy metrizable space. Theorem 2.6. Alexadroff) A fuzzy metrizable space. G set i a complete fuzzy metric space is a topologically complete Proof. Let X,M, *) be a complete fuzzy metric space ad G be a where each G set i X, that is G G, G is ope i X. By the above theorem, there exists a complete fuzzy metric m o ad we may assume that m is bouded by ). Let H be the Cartesia product the product topology. The H is a complete fuzzy metrizable space. Now, for each f G G G with let : G be the iclusio map. So the evaluatio map e : G H is a embeddig. Image of e is the diagoal G which is a closed subset of H ad by Corollary.6, G is complete. Thus G is a complete fuzzy metrizable space ad so is G which is homeomorphic to it. Refereces [] M.Amii ad R. Saadati, Topics i fuzzy metric space, J. Fuzzy Math., 423) 765-768. [2] A. George ad P.Veeramai, o some result i fuzzy metric space, fuzzy Sets ad System, 64 994) 395-399. [3] A. George ad P. Veeramai, O some result of aalysis for fuzzy metric spaces, Fuzzy Sets ad Systems, 9997)365-368. [4] V.Gregori ad S.Romaguera, Some properties of fuzzy metric spaces, Fuzzy Sets ad Systems, 52)485-489. [5] V. Gregori ad S. Romaguera, O completio of fuzzy metric spaces, Fuzzy Sets ad Systems, 3 22)399-44. [6] V. Gregori ad S. Romaguera, Characterizig completable fuzzy metric spaces, Fuzzy Sets ad Systems, 4424) 4-42. 48
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