Some notes on induced functions and group actions on hyperspaces

TL;DR

This paper investigates conditions under which continuous maps and group actions on topological spaces induce continuous functions and actions on their hyperspaces, with characterizations for Fell and Attouch-Wets hyperspaces.

Contribution

It provides new characterizations of when induced functions and group actions on hyperspaces are continuous, focusing on Fell and Attouch-Wets hyperspaces.

Findings

Characterized continuity of induced functions for specific hyperspaces

Identified conditions for continuous group actions on hyperspaces

Extended understanding of hyperspace topologies and their relation to original spaces

Abstract

We discuss the problem of when a continuous map between topological spaces induces a continuous function between their respective hyperspaces. We characterize the continuity of the induced function in the case of the Fell and Attouch-Wets hyperspaces. Additionally we explore the problem of whether a continuous action of a topological group $G$ on a topological space $X$ induces a continuous action on $CL(X)$.