On Topology of Fuzzy Strong b-Metric Spaces

Year 2018, Issue: 21, 59 - 67, 27.02.2018

Tarkan Oner

Abstract

In this study, we introduce and investigate the concept of fuzzy strong b-metric space such that is a fuzzy analogy of strong b-metric spaces. By using the open balls, we define a topology on these spaces which is Hausdor® and first countable. Later we show that open balls are open and closed balls are closed. After defining the standard fuzzy strong b-metric space induced by a strong b-metric, we show that these spaces have same topology. We also note that every separable fuzzy strong b-metric space is second countable. Moreover, we give the uniform convergence theorem for these spaces.

Keywords

Fuzzy strong b-metric space, strong b-metric space, b-metric spaces, uniform convergence

References

• [1] A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems 64 (1994) 395-399.
• [2] B. Schweizer, A. Sklar, Statistical metric spaces, Pasi¯c J. Maths. 10 (1960) 314-334.
• [3] I.A. Bakhtin, The contraction mapping principle in quasimetric spaces (Russian), Func An. Gos Ped Inst Unianowsk 30 (1989) 26-37.
• [4] J. Heinonen, Lectures on analysis on metric spaces, Springer Science & Business Media, 2012.
• [5] J. L. Kelley, General Topology, Springer Science & Business Media, 1975.
• [6] L. A. Zadeh, Fuzzy sets, Inform. and Control. 8 (1965) 338-353.
• [7] M. A. Erceg, Metric spaces in fuzzy set theory, J. Math. Anal. Appl. 69 (1979) 205-230.
• [8] M. A. Khamsi, N. Hussain,KKM mappings in metric type spaces, Nonlinear Anal. 73 (2010) 3123-3129.
• [9] O. Kramosil, J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975) 326-334.
• [10] O. Kaleva, S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems 12 (1984) 215-229.
• [11] P. Kumam, N.V. Dung, V.T.L. Hang, Some equivalences between cone b-metric spaces and b-metric spaces, Abstr. Appl.Anal. 2013 (2013) 1-8.
• [12] R. Saadati, On the Topology of Fuzzy Metric Type Spaces, Filomat 29:1 (2015) 133-141.
• [13] R. Fagin, L. Stockmeyer, Relaxing the triangle inequality in pattern matching, Int. J. Comput. Vis. 30(3) (1998) 219-231.
• [14] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math Inform Univ Ostraviensis 1(1) (1993) 5-11.
• [15] T. V. An, L. Q. Tuyen, N.V. Dung, Stone-type theorem on b-metric spaces and applications, Topology Appl. 185-186 (2015) 50-64.
• [16] W. Kirk, N. Shahzad, Fixed point theory in distance spaces, Springer, 2014.
• [17] Z. Deng, Fuzzy pseudo metric spaces, J. Math. Anal. Appl. 86 (1982) 74-95.