Some topological properties of fuzzy strong b-metric spaces

Subject Areas : General topology

T. Oner ¹

1 - Department of Mathematics, Faculty of Science, Mugla Sıtkı Kocman University Mugla 48000, Turkey

Received: 2018-08-28 Accepted : 2019-01-22 Published : 2019-06-01

Keywords: Fuzzy strong b-metric space, strong b-metric space, complete, separable,

Abstract :

‎In this study‎, ‎we investigate topological properties of fuzzy strong‎b-metric spaces defined in [13]‎. ‎Firstly‎, ‎we prove Baire's theorem for‎‎these spaces‎. ‎Then we define the product of two fuzzy strong b-metric spaces‎‎defined with same continuous t-norms and show that $X_{1}\times X_{2}$ is a‎‎complete fuzzy strong b-metric space if and only if $X_{1}$ and $X_{2}$ are‎‎complete fuzzy strong b-metric spaces‎. ‎Finally it is proven that a subspace‎‎of a separable fuzzy strong b-metric space is separable‎.

References:

[1] 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.

[2] I. A. Bakhtin, The contraction mapping principle in quasimetric spaces, Func An. Gos. Ped. Inst. Unianowsk. 30 (1989), 26-37.

[3] S. Czerwik, Contraction mappings in b-metric spaces, Acta. Math. Inform. Univ. Ostra. 1 (1) (1993), 5-11.

[4] Z. Deng, Fuzzy pseudo metric spaces, J. Math. Anal. Appl. 86 (1982), 74-95.

[5] M. A. Erceg, Metric spaces in fuzzy set theory, J. Math. Anal. Appl. 69 (1979), 205-230.

[6] R. Fagin, L. Stockmeyer, Relaxing the triangle inequality in pattern matching. Int. J. Comput. Vis. 30 (3) (1998), 219-231.

[7] A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems. 64 (1994), 395-399.

[8] J. Heinonen, Lectures on Analysis on Metric Spaces. Springer Science & Business Media, 2012.

[9] O. Kaleva, S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems. 12 (1984), 215-229.

[10] M. A. Khamsi, N. Hussain, KKM mappings in metric type spaces, Nonlinear Anal. 73 (2010), 3123-3129.

[11] W. Kirk, N. Shahzad, Fixed Point Theory in Distance Spaces, Springer, 2014.

[12] O. Kramosil, J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica, 11 (1975), 326-334.

[13] T.Öner, On topology of fuzzy strong b-metric spaces, J. New Theory. 21 (2018), 59-67.

[14] B. Schweizer, A. Sklar, Statistical metric spaces, Pasific J. Maths. 10 (1960), 314-334.

[15] L. A. Zadeh, Fuzzy sets, Inform. and Control. 8 (1965), 338-353.