## Abstract

We prove that every 3-generalized metric space is metrizable. We also show that for any ʋ with ʋ ≥ 4, not every ʋ-generalized metric space has a compatible symmetric topology.

Show Summary Details# Only 3-generalized metric spaces
have a compatible symmetric topology

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*Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas*, 2018*Journal of Inequalities and Applications*, 2017, Volume 2017, Number 1*Fixed Point Theory and Applications*, 2017, Volume 2017, Number 1*Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas*, 2017*Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas*, 2017*Journal of Inequalities and Applications*, 2016, Volume 2016, Number 1

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Tomonari Suzuki / Badriah Alamri / Misako Kikkawa

We prove that every 3-generalized metric space is metrizable. We also show that for any ʋ with ʋ ≥ 4, not every ʋ-generalized metric space has a compatible symmetric topology.

Keywords: ʋ-generalized metric space; Metrizability; Topology; Symmetrizable; Semimetrizable

[1] B. Alamri, T. Suzuki and L. A. Khan, Caristi’s fixed point theorem and Subrahmanyam’s fixed point theorem in ʋ-generalized metric spaces, J. Funct. Spaces, 2015, Art. ID 709391, 6 pp. Web of ScienceGoogle Scholar

[2] A. Branciari, A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces, Publ. Math. Debrecen, 57 (2000), 31–37. MR1771669 Google Scholar

[3] G. Gruenhage, “Generalized metric spaces” in Handbook of set-theoretic topology, 1984, pp. 423–501, North-Holland, Amsterdam. MR0776629 Google Scholar

[4] Z. Kadelburg and S. Radenovi´c, On generalized metric spaces: A survey, TWMS J. Pure Appl. Math., 5 (2014), 3–13. Google Scholar

[5] W. A. Kirk and N. Shahzad, Generalized metrics and Caristi’s theorem, Fixed Point Theory Appl., 2013, 2013:129. MR3068651 Web of ScienceGoogle Scholar

[6] T. Suzuki, Generalized metric spaces do not have the compatible topology, Abstr. Appl. Anal., 2014, Art. ID 458098, 5 pp. Web of ScienceGoogle Scholar

[7] S. Willard, General Topology, Dover (2004). MR2048350 Google Scholar

**Received**: 2015-01-27

**Accepted**: 2015-08-03

**Published Online**: 2015-09-04

**Citation Information: **Open Mathematics, Volume 13, Issue 1, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2015-0048.

©2015 Tomonari Suzuki et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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