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Ultradiversities and their spherical completeness

Mehrabani Gholamreza H., Nourouzi Kourosh

Journal of Applied Analysis

2020

Vol. 26, nr 2

231--240

Diversities are a generalization of metric spaces which associate a positive real number to every finite subset of the space. In this paper, we introduce ultradiversities which are themselves simultaneously diversities and a sort of generalization of ultrametric spaces. We also give the notion of spherical completeness for ultradiversities based on the balls defined in such spaces. In particular, with the help of nonexpansive mappings defined between ultradiversities, we show that an ultradiversity is spherically complete if and only if it is injective.

The Mackey topolgy for locally convex modules over a valuation ring

Kubzdela A.

Bulletin of the Polish Academy of Sciences. Mathematics

1999

Vol. 47, no 1

27-36

In this paper we study the duality for locally convex modules over the unit disk of a spherically complete valued field. We consider a dual pair for arbitrary locally convex B[K]-modules and pairs consisting of a locally convex module and its dual. We prove that the Mackey topology for an arbitrary locally convex B[K]-module exists. This extends some results of Van Tiel [1] obtained for locally convex spaces. Some additional facts for compatible topologies on c-compact B[K]-modules are included.