Journal of Convex Analysis 29 (2022), No. 3, 717--730
Copyright Heldermann Verlag 2022
Metrization of Idempotent Convex Compacta
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
[Abstract-pdf]
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 $A$ of an $I$-convex compactum $X$ to its closed idempotent convex hull is continuous. This implies that each neighborhood of the diagonal $\Delta_X\subset X\times X$ contains an idempotent convex neighborhood. The main result is the theorem that the topology on an idempotent convex compactum $X$ is determined by a family of idempotent convex pseudometrics (with one idempotent convex metric if $X$ is metrizable).
Keywords: I-convex compactum, idempotent semimodule, locally convex space, Vietoris topology, metrization
MSC: 52A01, 52A30, 15A80, 46S99.
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