Vietoris-type Topologies on Hyperspaces

TL;DR

This paper introduces a new Vietoris-type hypertopology on hyperspaces, compares it with existing topologies, and extends classical results to this new setting, enriching the theory of hyperspace topologies.

Contribution

It defines a novel Vietoris-type hypertopology, analyzes its properties, and extends classical hyperspace results to this new topology, highlighting differences from the traditional Vietoris topology.

Findings

The new Vietoris-type hypertopology generally differs from the classical Vietoris topology.

Some classical results about hyperspaces with Vietoris topology are extended to the new topology.

The paper identifies conditions under which the new topology coincides or differs from existing topologies.

Abstract

We introduce a new Vietoris-type hypertopology by means of the upper-Vietoris-type hypertopology defined by G. Dimov and D. Vakarelov [On Scott consequence systems, Fundamenta Informaticae, 33 (1998), 43-70] (it was called there {\em Tychonoff-type hypertopology}) and the lower-Vietoris-type hypertopology introduced by E. Ivanova-Dimova [Lower-Vietoris-type topologies on hyperspaces, Topology Appl. (to appear)]. We study this new Vietoris-type hypertopology and show that it is, in general, different from the Vietoris topology. Also, some of the results of E. Michael [Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182] about hyperspaces with Vietoris topology are extended to analogous results for hyperspaces with Vietoris-type topology. We obtain as well some results about hyperspaces with Vietoris-type topology which concern some problems analogous to those regarded by H.-J. Schmidt [Hyperspaces of quotient and subspaces. I. Hausdorff topological spaces, Math. Nachr. 104 (1981), 271-280].