Metric and topology on the poset of compact pseudoultrametrics

Authors

  • S. Nykorovych Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine
  • O. Nykyforchyn Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine; Casimir the Great University in Bydgoszcz, 30 J.K. Chodkiewicza str., 85064, Bydgoszcz, Poland
https://doi.org/10.15330/cmp.15.2.321-330

Keywords:

pseudoultrametric, metrization, сompactum, hypograph
Published online: 2023-08-03

Abstract

In two ways we introduce metrics on the set of all pseudoultrametrics, not exceeding a given compact pseudoultrametric on a fixed set, and prove that the obtained metrics are compact and topologically equivalent. To achieve this, we give a characterization of the sets being the hypographs of the mentioned pseudoultrametrics, and apply Hausdorff metric to their family. It is proved that the uniform convergence metric is a limit case of metrics defined via hypographs. It is shown that the set of all pseudoultrametrics, not exceeding a given compact pseudoultrametric, with the induced topology is a Lawson compact Hausdorff upper semilattice.

Article metrics
How to Cite
(1)
Nykorovych, S.; Nykyforchyn, O. Metric and Topology on the Poset of Compact Pseudoultrametrics. Carpathian Math. Publ. 2023, 15, 321-330.