### Journal Menu >> Applied Mathematics, 2013, 4, 1296-1300 http://dx.doi.org/10.4236/am.2013.49175 Published Online September 2013 (http://www.scirp.org/journal/am) Modular Spaces Topology Ahmed Hajji Laboratory of Mathematics, Computing and Application, Department of Mathematics, Faculty of Sciences, Mohammed V-Agdal University, Rabat, Morocco Email: hajid2@yahoo.fr Received April 29, 2013; revised May 29, 2013; accepted June 7, 2013 Copyright © 2013 Ahmed Hajji. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT In this paper, we present and discuss the topology of modular spaces using the filter base and we then characterize closed subsets as well as its regularity. Keywords: Topology of Modular Spaces; ∆2-Condition; Filter Base 1. Introduction In the theory of the modular spaces X, the notion of ∆2-condition depends on the convergence of the se-quences in modular space X. More precisely, it reads: for any sequence nnx in X, if m 20nnxli, we have 0nxlim 2n . This condition has been used to study the topology of modular spaces, see J. Musielak , and to establish some fixed point theorems in modu-lar spaces, see [2-7]. Some fixed point theorems without ∆2-condition can be found in [8,9]. In this paper, we present a new equivalent form for the ∆2-condition in the modular spaces X which is used to show that the corresponding topology is separate and to establish some associated topological properties, includ-ing the characterization of the -closed subsets as well as its regularity. The present work is an improved Eng-lish version of a pervious preprint in French . 2. Preliminaries We begin by recalling some definitions. Definition 2.1 Let X be an arbitrary vector space over K or . 1) A functional :0,X0x is called modular if 0x implies . a)  xx for any xX when K, and b) eitxx for any real t when K. c) xyx  y0 for , and 1. 2) If we replace c) by the following xyx  y0 for , and 1, then the modular  is called convex. 3) For given modular  in X, the 0X x 0 as Xx is called a modular space. 4) a) If  is a modular in X, then inf x0, xuuu  is a F-norm. b) Let  be a convex modular, then inf0, 1xxuu is called the Luxemburg norm. 3. Topology τ in Modular Spaces In this section, we introduce the property 0 for a mod-ular , which will be used to show that the corre-sponding topology, noted by , on modular space X is separate, and to characterize their closed subsets. We begin with the following Proposition 3.1 Consider the family 0Brr0,, where 0,BrxXxr. Then 1) The family is a filter base. 2) Any element of  is balanced and absorbing. Furthermore, if  is convex, then any element of is convex. Proof. 1) is a filter base. Indeed, we have Copyright © 2013 SciRes. AM A. HAJJI 1297a) because any 0,Br. b) Let and 10,Br0,Br2zB be in and set . Then, for any we have 12,rrinfr0, r212zrrzrr and therefore . That is  10, 0,zB rB r 120,0,0, .BrBrBr Hence is a filter base for the existence of 0,Br such that  120,0, 0,BrBrBr. 2) Let . 0,Br0,Bra) is balanced. Indeed, for given ei with  and 1, and given 0,xBr, we have  ei0rexists >0, such that 0< < and 0 such that 0,xBr. This shows that is absorbing. 0, rBNow, assume that  is in addition convex and let . For given 0,Br,0,xyB r and 0,1, we have   11<,xyx y  r,. then  10xyB r  Thence is convex. 0,BrDefinition 3.1 We say that  satisfies the property 0 if for all >0, there exist and >0L>0 such that  yx for every x, y satisfying xL and 0 such that  000,0,0, .BB B r In fact, let ; >>0r. Since  satisfies the property 0, there are and >0L>0, such that for 0 there exists such that 0N0,Bnxx whenever . 0Note that the property 0>nN is a necessary condition to show the uniqueness of the limit when exists. Thus, the -convergence need the property 0 and it is easy to see that -convergence and -convergence are equiva-lent. Definition 3.4 Let  be a modular satisfying the property 0. A subset B of X is said to be -losed if and only if the complimentary of B in cX, noted by BXC, is an element of . The following lemma shows that the property 0 makes sense in the theory of modular spaces. Lemma 3.1 Let  be a modular and X be a modular space. Then  satisfies the -condition if and only if 2 satisfies the property 0. Proof. To prove “if”, let nnx be a sequence in X such that as . This implies that for all 0nx>0n, there exists such that for any we have 0n0>nninf, ,.2nxL Now, take nnXx and , for any . It follows 2nYxn0nn  inf, ,.2nnnnXxYX L  This yields  22nn nYx x  when- ever 0. Whence, the sequence nn2nxn tends to zero as n goes to , and therefore  satisfies the 2-condition. For “only if”, let  be a modular satisfying the 2-condition, and suppose that there exists >0 such that for any and for any >0L>0, there exist ,xyX satisfying <,xL 0nnyx for any n. Finally, for all >0, there are and >0L>0 such that if x< and yx<, we have 0, suchthat,.FFXXFXxFxC CBxBBxBxFC  Copyright © 2013 SciRes. AM A. HAJJI 1299Finally, >0, ,.xFBx F  Proof of Theorem 3.2. Let F be -closed and n be a sequence in nxF such that nxx. Then, for any >0, there exists 0 such that for every , we have n0>nn,nxBx. This implies that >0, ,.BxF Whence, making use of Proposition 3.1, we get that xF. Conversely, assume that F is not -closed, then FXC is not an open set for the -topology. There exists then FXxC satisfying ,FXCBx and so ,Bx F for any >0. Therefore, for 1k there exists 1,kxBxkFnn. Thence, the obtained sequence xF satisfies nxx. This implies xF, which is in contradiction with the fact that FXxC. In conclusion, F is -closed. Remark 3.1 Observe that 20satisfiesthe-conditionsatisfies the property. As consequence, we see that under the assumption that  satisfies the 0 property, we have 1topology topology. Then definitions of -convergence and -closed subsets of X need the hypothesis that  satisfies the ∆2-condition. The following result shows that the modular space X is a regular space. Theorem 3.3 Let  be a modular satisfying the ∆2- condition, A be a -closed subset of X and 0xA. Then there exists an open neighborhood 0xV of 0x such that . 0xIn order to show the theorem above, we need the following result. VAProposition 3.3 Let  be a modular satisfying the ∆2-condition and AX. Then  ,inf ,xAx y yA0 if and only if xA, where A is the closure of A for the -topology. Proof. We have  ,inf ,xAx y yA0. Then for any 1n, there exists such that nyA1r0. Next, since  satisfies the 2-condition then by Lemma 3.1, for >03r, there exist , and 0L0 such that if xL and yx we have  yx . More- over, there exists such that *0minf ,rLm whenever 0. Now, let and we consider the open neighborhood of mm1max 3,m00mx 0010, .xrVxB m Suppose next that 0 and let 0xVAxyVA. Since A is closed we make use of Proposition 3.1 to exhibit a sequence nnyAy such that . So nythat one considers nnXyy and . Since 0nYxynnyA and 0xA, then . On the other hand, note that nYr 1inf, ,nnrXyy Lm whenever and 0nn01inf, .nn rXY xyLm  Therefore  1233nnrr rrY yym  whenever , a contradiction. Thus 0nn0xVA. Remark 3.2 If  satisfies Fa tou p roperty, then  0,Br 0,B rxXxr  is a closed ball of the topology . We note by ,fBxr0r all closed ball centered at x with the radius (see ). Corolla ry 3.1 Under th e same hypotheses of Theorem Copyright © 2013 SciRes. AM A. HAJJI Copyright © 2013 SciRes. AM 1300 3.3, and if the modular  satisfies Fatou property, then By Proposition 3.1, there exists 10,nnryBm such that . Moreover, the sequence ny0xVA. Proof. Making appeal of Theorem 3.3, there exists y00nn0010,xrVxBm such that . Then, we have 0xVA0010,xfrVxBmxxyV satisfying 00nxyxy. Hence . Indeed, let 0xyV and note that from Proposition 3.1, there exists a sequence 10,nfnryBm00.xxyV Finally, we take the same arguments as in the proof of Theorem 3.3, we have  such that 0.xVA 0,nxyy REFERENCES which implies that . Indeed, it is easy to see that and since 0nyyx00x nnYy y satisfies the -condition we have also . 202Xyyx J. Musielak, “Orlicz Spaces and Modular Spaces,” Lec-ture Notes in Mathematics, Vol. 1034, 1983. 0nn A. Ait Taleb and E. Hanebaly, “A Fixed Point Theorem and Its Application to Integral Equations in Modular Function Spaces,” Proceedings of the American Mathe- matical Society, Vol. 128, 2000, pp. 419-426. doi:10.1090/S0002-9939-99-05546-X Thence, for 0, there are and 0L0 such that  A. Razani and R. Moradi, “Common Fixed Point Theo- rems of Integral Type in Modular Spaces,” Bulletin of the Iranian Mathematical Society, Vol. 35, No. 2, 2009, pp. 11-24. inf, ,,2nXL and  A. Razani, E. Nabizadeh, M. B. Mohammadi and S. H. Pour, “Fixed Point of Nonlinear and Asymptotic Contrac- tions in the Modular Space,” Abstract and Applied Analy- sis, Vol. 2007, 2007, Article ID: 40575. inf, ,,2nnnYX YL   whenever , then 0nn A. P. Farajzadeh, M. B. Mohammadi and M. A. Noor, “Fixed Point Theorems in Modular Spaces,” Mathemati- cal Communications, Vol. 16, 2011, pp. 13-20.  0inf, ,,2222nnYyyxL   M. A. Khamsi, “Nonlinear Semigroups in Modular Func-tion Spaces,” Thèse d'état, Département de Mathé-matiques, Rabat, 1994. whenever . Therefore 0nn0110,0,.nfrryyxB Bmm      M. A. Khamsi, W. Kozlowski and S. M.-Reich, “Fixed Point Theory in Modular Function Spaces,” Nonlinear Analysis, Theory, Methods and Applications, Vol. 14, No. 11, 1990, pp. 935-953.  F. Lael and K. Nourouzi, “On the Fixed Points of Corre- spondences in Modular Spaces,” ISRN Geometry, Vol. 2011, 2011, Article ID: 530254. doi:10.5402/2011/530254 It follows 00010, ,fryxyxx Bm   M. A. Khamsi, “Quasicontraction Mappings in Modular Spaces without ∆2-Condition,” Fixed Point Theory and Applications, Vol. 2008, 2008, Article ID: 916187. and hence  A. Hajji, “Forme Equivalente à la Condition ∆2 et Cer-tains Résultats de Séparations dans les Espaces Modu-laires,” 2005. http://arXiv.org/abs/math.FA/0509482 0010,.xfrVxBm Inversely, let 0010, .frxyxB m 