Jardon, D., Sanchis, M. (2018). POINTWISE CONVERGENCE TOPOLOGY AND FUNCTION SPACES IN FUZZY ANALYSIS. Iranian Journal of Fuzzy Systems, 15(2), 1-21. doi: 10.22111/ijfs.2018.3753

D. R. Jardon; M. Sanchis. "POINTWISE CONVERGENCE TOPOLOGY AND FUNCTION SPACES IN FUZZY ANALYSIS". Iranian Journal of Fuzzy Systems, 15, 2, 2018, 1-21. doi: 10.22111/ijfs.2018.3753

Jardon, D., Sanchis, M. (2018). 'POINTWISE CONVERGENCE TOPOLOGY AND FUNCTION SPACES IN FUZZY ANALYSIS', Iranian Journal of Fuzzy Systems, 15(2), pp. 1-21. doi: 10.22111/ijfs.2018.3753

Jardon, D., Sanchis, M. POINTWISE CONVERGENCE TOPOLOGY AND FUNCTION SPACES IN FUZZY ANALYSIS. Iranian Journal of Fuzzy Systems, 2018; 15(2): 1-21. doi: 10.22111/ijfs.2018.3753

POINTWISE CONVERGENCE TOPOLOGY AND FUNCTION SPACES IN FUZZY ANALYSIS

^{1}Academia de Matematicas, Universidad Autonoma de la Ciudad de Mexico, Calz. Ermita Iztapalapa s/n, Col. Lomas de Zaragoza 09620, Ciudad de Mexico , Mexico

^{2}Institut de Matematiques i Aplicacions de Castello (IMAC), Universitat Jaume I, Campus Riu Sec, 12071-Castello, Spain

Abstract

We study the space of all continuous fuzzy-valued functions from a space $X$ into the space of fuzzy numbers $(\mathbb{E}\sp{1},d\sb{\infty})$ endowed with the pointwise convergence topology. Our results generalize the classical ones for continuous real-valued functions. The field of applications of this approach seems to be large, since the classical case allows many known devices to be fitted to general topology, functional analysis, coding theory, Boolean rings, etc.

[1] R. F. Arens, A topology for spaces of transformations, Ann. of Math., 47(2) (1946), 480{495. [2] A. V. Arkhangel'ski, Topological Function Spaces, Translated from the Russian by R. A. M. Hoksbergen, Mathematics and its Applications (Soviet Series), 78, Kluwer Academic Publishers Group, Dordrecht, 1992. [3] T. Berger and L. D. Davisson, Advances in Source Coding, International Centre for Mechanical Sciences (CISM) Courses and Lectures, No. 166, Springer-Verlag, Vienna-New York, 1975.

[4] G. Bosi, J. C. Candeal, E. Indurain, E. Oloriz and M. Zudaire, Numerical representations of interval orders, Order 18(2) (2001), 171{190. [5] D. S. Bridges and G. B. Mehta, Representations of Preferences Orderings, Lecture Notes in Economics and Mathematical Systems, 422, Springer-Verlag, Berlin, 1995. [6] M. M. Choban and M. I. Ursul, Applications of the Stone Duality in the Theory of Precompact Boolean Rings, Advances in ring theory, 85{111, Trends Math., Birkhauser/Springer Basel AG, Basel, 2010. [7] L. Dengfeng, Properties of b-vex fuzzy mappings and applications to fuzzy optimization, Fuzzy Sets and Systems, 94 (1998), 253{260. [8] S. Dey, T. Mukhopadhyay, H. H. Khodaparast and S. Adhikari, Fuzzy uncertainty propagation in composites using Gram-Schmidt polynomial chaos expansion, Appl. Math. Model., 40(7{ 8) (2016), 4412{4428. [9] P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets. Theory and Applications, World Scientic Publishing Co., Inc., River Edge, NJ, 1994. [10] D. Dubois and H. Prade, Operations on fuzzy numbers, Internat. J. Systems Sci., 9(6) (1978), 613{626. [11] R. Engelking, General Topology, Translated from the Polish by the author. Second edition, Sigma Series in Pure Mathematics, 6, Heldermann Verlag, Berlin, 1989. [12] J. J. Font, A. Miralles and M. Sanchis, On the fuzzy number space with the level convergence topology, J. Funct. Spaces Appl., 2012, Art. ID 326417, 11 pp. [13] J. J. Font and M. Sanchis, Sequentially compact subsets and monotone functions: an appli- cation to fuzzy theory, Topology Appl., 192 (2015), 113{122. [14] A. Garca-Maynez and S. Romaguera, Perfect pre-images of conally complete metric spaces, Comment. Math. Univ. Carolin., 40(2) (1999), 335{342. [15] R. Jr. Goetschel and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18(1) (1986), 31{43. [16] J. Nagata, On lattices of functions on topological spaces and of functions on uniform spaces, Osaka Math. J., 1 (1949), 166{181. [17] M. S. Osborne, Locally Convex Spaces, Graduate Texts in Mathematics, 269, Springer, Cham, 2014. [18] G. Picci and D. S. Gilliam, Dynamical systems, control, coding, computer vision. New trends, interfaces, and interplay, Papers from the Mathematical Theory of Networks and Systems Symposium (MTNS-98) held in Padova, July 6{10, 1998. Edited by Giorgio Picci and David S. Gilliam. Progress in Systems and Control Theory, 25. Birkhauser Verlag, Basel, 1999. [19] V. I. Ponomarev and V. V. Tkachuk, The countable character of X in X versus the countable character of the diagonal in X X, Vestnik Moskov. Univ. Ser. I Mat. Mekh., (in Russian), 104(5) (1987), 16{19. [20] X. Ren and Ch. Wu, The fuzzy Riemann-Stieltjes integral, Internat. J. Theoret. Phys., 52(6) (2013), 2134{2151. [21] S. Romaguera, On conally complete metric spaces, Questions Answers Gen. Topology, 16 (1998), 165{169. [22] L. Stefanini and B. Bede, Generalized fuzzy dierentiability with LU-parametric representa- tion, Fuzzy Sets and Systems, 257 (2014), 184{203. [23] V. V. Tkachuk, A Cp-theory Problem Book. Topological and Function Spaces, Problem Books in Mathematics, Springer, New York, 2011. [24] V. V. Tkachuk, A Cp{theory Problem Book. Special Features of Function Spaces, Problem Books in Mathematics, Springer, Cham, 2014. [25] C. Wu, Function spaces and application to fuzzy analysis, Function Spaces VIII, Banach Center Publ., Polish Acad. Sci. Inst. Math., Warsaw, 79 (2008), 235{246. [26] J. F. F. Yao and J. S. Yao, Fuzzy decision making for medical diagnosis based on fuzzy number and compositional rule of inference, Fuzzy Sets and Systems, 120 (2001), 351{366. [27] G. Zhang, Y. H.Wu, M. Remias and J. Lu, Formulation of fuzzy linear programming problems as four-objective constrained optimization problems, Appl. Math. Comput., 139(2{3) (2003), 383{399.