On $b_2$-Metric Spaces

Abstract

The target of this paper is to induce a topology from a given $b_2$-metric and study the properties of the topology induced by this way. We first define the notion of $\varepsilon$-ball in $b_2$-metric spaces and consider the topology induced by a given $b_2$-metric via $\varepsilon$-balls. We study some properties of this topological space such as separation axioms and semi-metrizability. Also, we show with the examples that some known properties for $\varepsilon$-balls in metric spaces have not existed in $b_2$-metric spaces. Then we introduce the concept of strong $b_2$-metric spaces in which these known properties are provided. Finally, we show that every strong $b_2$-metric topological space is normal, metrizable and of second category.

Keywords

• b_2-metric spaces
• strong b_2-metric spaces
• Hausdorff Spaces
• Metrizable topological spaces

References

1. [1] A. Aliouche, C.Simpson, Fixed Points and Lines in 2-Metric Spaces, Advances In Math. Vol: 229 (2012) 668-690.
2. [2] I. A. Bakhtin, The contraction mapping principle in quasimetric spaces, Funct. Anal. Unianowsk Gos. Ped. Inst. Vol: 30 (1989), 26-37.
3. [3] S. Czerwik, Contraction mappings in b-metric spaces, Acta Mathematica Et Informatica Universitatis Ostraviensis Vol: 1 (1993), 5-11.
4. [4] M. R. Farhangdoost, Metrizable and 2-Metrizable Topological Spaces, Journal of Dynamical Systems and Geometric Theories Vol: 10 (2012), 61-69.
5. [5] S. Gahler,¨ 2-Metrische Raume¨ und ihre topologische struktur, Math. Nachr. Vol:26 (1963), 115-118.
6. [6] S. Gahler,¨ Lineare 2-normierte Raume,¨ Math. Nachr. Vol: 28 (1965), 1-43.
7. [7] S. Gahler,¨ W. Gahler,¨ Espaces 2-Metriques Et Localement 2-Metriques, Ann.Scient.Ec.Norm.Sup. Vol: 3 (1965), 387-395
8. [8] K. Iseki, Fixed point theorems in 2-metric spaces, Math. Seminar Notes, Kobe Univ. Vol: 3 (1975) 133-136.

9. [9] M. A. Khamsi, N. Hussain, KKM mappings in metric type spaces, Nonlinear Analysis: Theory, Methods and Applications Vol: 73 No: 9 (2010), 3123-3129.
10. [10] W. Kirk, N. Shahzad, Fixed Point Theory in Distance Spaces, Springer, Cham (2014).
11. [11] B. K. Lahiri, P.Das, L.K.Dey, Cantor’s theorem in 2-metric spaces and its applications to fixed point problems, Taiwanese J.Math. Vol: 15 (2011) 337-352.
12. [12] Z. Mustafa, V. Parvaech, J. R. Roshan, Z. Kadelburg, b2-Metric Spaces and Some Fixed Point Theorems, Fixed Point Theory and Applications Vol: 144 (2014).
13. [13] S. V. R. Naidu, J. Rajendra Prasad, Fixed point theorems in 2-metric space, Indian J.Pure Appl.Math. Vol: 17 No: 8 (1986) 974-993.
14. [14] T. Van An, L. Q. Tuyen, N. Van Dung, Stone-type theorem on b-metric spaces and applications, Topology and its Applications Vol: 185 (2015) 50-64.