Hacettepe Journal of Mathematics and Statistics 2651-477X 2651-477X Hacettepe University Statistics İstatistik Relative Metric Spaces Relative Metric Spaces Malziri M. Molaei M.r. 05 01 2011 40 5 703 709 05 11 2014 Copyright © 2002, Hacettepe Journal of Mathematics and Statistics 2002 Hacettepe Journal of Mathematics and Statistics

In this paper the notion of a relative metric space, as a mathematical model compatible with a physical phenomena, is considered. The notion of relative topological entropy for relative semi-dynamical systems on a relative metric space is studied. It is proved that observational topological entropy is an invariant object up to a relative conjugate relation.

 Observer Relative structure Molecular lattice Relative topology Observational topological entropy 2000 AMS Classification: 37 B 40 George, A. and Veeramani, P. On some results of analysis for fuzzy metric spaces, Fuzzy Sets and Systems 90, 365–368, 1997. Liu, B. A survay of entropy of fuzzy varables, Journal of Uncertain Systems 1 (1), 4–13, Liu, B. Some research problems in uncertainty theory, Journal of Uncertain Systems, 3 (1), –10, 2009. Molaei, M. R. Observational modeling of topological spaces, Chaos, Solitons and Fractals 42, –619, 2009. Molaei, M. R. Mathematical modeling of observers in physical systems, Journal of Dynamical Systems and Geometric Theories 4 (2), 183–186, 2006. Molaei, M. R. Relative semi-dynamical systems, International Journal of Uncertainty, Fuzzi- ness and Knowledge-Based Systems, 12 (2), 237–243, 2004. Molaei, M. R. and Hoseini Anvari, M. R. Relative manifolds, Intelligent Automation and Soft Computing 14 (2), 219–226, 2008. Molaei, M. R., Hoseini Anvari, M. R. and Haqiri, T. On relative semi-dynamical systems, Intelligent Automation and Soft Computing, 13 (4), 405–413, 2007. Petr, H. Metamathematics of Fuzzy Logic (Dordrecht: Kluwer, ISBN 0792352389, 1998). Rahmat, M. R. S. and Noorani, M. S. M., Product of fuzzy metric spaces and fixed point theorems, Int. J. Contemp. Math. Sciences 3 (15), 703–712, 2008. Ramadan, A. A. and Abd El-latif, A. A. Supra fuzzy convergence of fuzzy filters, Bull. Korean Math. Soc. 45, 207–220, 2008. Rao, K. P. R., Babu, G. R. and Fisher, B. Common fixed point theorems in fuzzy metric spaces under implicit relations, Hacet. J. Math. Stat. 37 (2), 97–106, 2008. Ray, A. D. and Saha, P. K. Fixed point theorems on generalized fuzzy metric spaces, Hacet. J. Math. Stat. 39 (1), 1–9, 2010. Sostak, A. Two decades of fuzzy topology: Basic ideas, notions and results, Russian Math- ematical Surveys 44 (6), 125–186, 1989. Wang, G. J. Theory of topological molecular lattices, Fuzzy Sets and Systems 47 (3), 351– , 1992. Zadeh, L. A. Fuzzy sets, Inform. and Control 8, 338–353, 1965.