Lower-Vietoris-type Topologies on Hyperspaces

TL;DR

This paper introduces a new lower-Vietoris-type hypertopology on hyperspaces, generalizes existing results, and demonstrates the existence of invariant subcontinua under continuous maps on continua.

Contribution

It develops a novel lower-Vietoris-type hypertopology and extends previous results, including the existence of invariant subcontinua for continuous maps on continua.

Findings

New lower-Vietoris-type hypertopology introduced

Generalization of previous hyperspace topologies results

Existence of invariant subcontinua under continuous maps

Abstract

We introduce a new lower-Vietoris-type hypertopology in a way similar to that with which a new upper-Vietoris-type hypertopology was introduced in G. Dimov and D. Vakarelov, "On Scott consequence systems", Fundamenta Informaticae, 33 (1998), 43-70. (it was called there {\em Tychonoff-type hypertopology}). We study this new hypertopology and, in particular, we generalize many results from E. Cuchillo-Ibanez, M. A. Moron and F. R. Ruiz del Portal, "Lower semifinite topology in hyperspaces", Topology Proceedings, 17 (1992), 29-39. As a corollary, we get that for every continuous map $f:X\longrightarrow X$, where $X$ is a continuum, there exist a subcontinuum $K$ of $X$ such that $f(K)=K.$