On the extension of partial metrics and quasi-metrics

In a recent article, V. Mykhaylyuk and V. Myronyk investigated the problem of extending partial metrics and quasi-metrics. Concretely, they stated, and proved, as a main result that if A is a closed subset of a partially metrizable space (X,τ) then every compatible partial metric p on A whose induced quasi-metric q_p is bounded admits an extension to a compatible partial metric on X. They also gave an example showing that the boundedness of the quasi-metric q_p cannot be omitted. Since the partially metrizable space of their example is not T₁, the authors raised the natural question of whether the boundedness of the quasi-metric q_p can be removed when the partially metrizable space is T₁ . Moreover, they also posed a similar question in the framework of quasi-metrizable spaces. In this note we present two examples showing that Mykhaylyuk and Myronyk’s questions have a negative answer.

Show more Show less

References:

S. Assaf and T. Cuchta, The rational sequence is partially metrizable, Topol. Proc. 62 (2023), 65-72.

R. H. Bing, Extending a metric, Duke Math. J. 14 (1947), 511-519.https://doi.org/10.1215/S0012-7094-47-01442-7

J. Deák, Extending a quasi-metric, Studia Sci. Math. Hungar. 28 (1993), 105-113.

F. Hausdorff, Erweiterung einer stetigen Abbildung, Fund. Math. 30 (1938), 40-47. https://doi.org/10.4064/fm-30-1-40-47

E. Karapinar and R. P. Agarwal, Fixed Point Theory in Generalized Metric Spaces, Springer (2023). https://doi.org/10.1007/978-3-031-14969-6

H. Lu, H. Zhang, and W. He, Some remarks on partial metric spaces, Bull. Malays. Math. Sci. Soc. 43 (2020), 3065-3081. https://doi.org/10.1007/s40840-019-00854-1

S. G. Matthews, Partial metric topology, Proc. 8th Summer Conference on General Topology and Applications, Ann. New York Acad. Sci. 728 (1994), 183-197.https://doi.org/10.1111/j.1749-6632.1994.tb44144.x

S. Matthews and H. Pajoohesh, Infinity valued partial metrics, Appl. Gen. Topol. 26 (2025), 71-85.https://doi.org/10.4995/agt.2025.20840

V. Mykhaylyuk and V. Myronyk, Compactness and complementness in partial metric spaces, Topology Appl. 270 (2020), 106925.https://doi.org/10.1016/j.topol.2019.106925

V. Mykhaylyuk and V. Myronyk, Metrizability of partial metric spaces, Topology Appl. 308 (2022), 107949.https://doi.org/10.1016/j.topol.2021.107949

V. Mykhaylyuk and V. Myronyk, Extending of partial metrics, Carpathian Math. Publ. 17 (2025), 33-41.https://doi.org/10.15330/cmp.17.1.33-41

H. Pajoohesh, $T_{0}$ functional Alexandroff topologies are partial metrizable, Appl. Gen. Topol. 25 (2024), 305-319.https://doi.org/10.4995/agt.2024.19401

S. Romaguera, P. Tirado, and O. Valero, Complete partial metric spaces have partially metrizable computational models, Int. J. Comput. Math. 89 (2012), 284-290.https://doi.org/10.1080/00207160.2011.559229

P. Waszkiewicz, Partial metrisability of continuous posets, Math. Struct. Comput. Sci. 16 (2006), 359-372.https://doi.org/10.1017/S0960129506005196

Show more Show less