A Note on Quasi-Metrizable Spaces

Abstract

In a previous paper on quasi-uniformizable spaces related to statistical metric spaces it was determined three conditions to obtain a first countable T1-topology. In this paper, we determine two conditions to define a quasi-metrizable topological space. Namely, we determine the conditions to obtain both a quasi-metric and a quasi-uniform topology coincides with a quasi-metrizable topological space.

Keywords

• uniform space
• quasi-uniform space
• distribution function
• distance function
• statistical metric space

References

1. Csaszar, A., Fondements de la Topologie Ge´ne´rale, Akade´miai Kiado´, Budapest, 1960.
2. Duru, H., Ilter, S., Bilgin, A., A first countable T1 topology as related to statistical metric spaces, Yüzüncü Yıl Üniversitesi Fen Bilimleri Enstitüsü Dergisi, (in press), (2022).
3. Fletcher, P., William F.L., Lecture Notes in Pure and Applied Mathematics Quasi-uniform Spaces, CRC Press, 1982.
4. Kelley, J.L., General Topology, Springer, 1975.
5. Nachbin, L., Sur les espaces uniformes ordonn´es, Comptes Rendus Hebdomadaires des S´eances de l’Aca´emie des Sciences,226(10)(1948), 774–775.
6. Pervin, W.J., Quasi-uniformization of topological spaces, Math. Annalen 147(1962), 316–317.
7. Schweizer, B., Sklar, A., Thorp, E., The metrization of statistical metric spaces, Pacific Journal of Mathematics, 10(1960), 673–675.