Abstract

Using a convenient subbase on the second hyperspace of a compactum with the Vietoris topology, we prove that the mapping that takes each closed non-empty subset AA of an II-convex compactum XX to its closed idempotent convex hull is continuous. This implies that each neighborhood of the diagonal ΔX⊂X×X\Delta_X\subset X\times X contains an idempotent convex neighborhood. The main result is the theorem that the topology on an idempotent convex compactum XX is determined by a family of idempotent convex pseudometrics (with one idempotent convex metric if XX is metrizable).

Author information

Contact details are reproduced from the original publication and may be historical.

Oleh Nykyforchyn

Institute of Mathematics, Casimir the Great University, Bydgoszcz, Poland
and: Dept. of Mathematics and Computer Science, V. Stefanyk Precarpathian National University, Ivano-Frankivsk, Ukraine

oleh.nyk@gmail.com

Mariia Savchyn

Dept. of Mathematics and Computer Science, V. Stefanyk Precarpathian National University, Ivano-Frankivsk, Ukraine

savchyn.mar@gmail.com

Keywords

  • I-convex compactum
  • idempotent semimodule
  • locally convex space
  • Vietoris topology
  • metrization

Mathematics Subject Classification

52A01, 52A30, 15A80, 46S99

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Suggested citation

O. Nykyforchyn, M. Savchyn. “Metrization of Idempotent Convex Compacta.” Journal of Convex Analysis 29 (2022), No. 3, 717–730. https://doi.org/10.68381/jca29040

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Published by Heldermann Verlag, 2022. Rights now held by Banach Press LLC.