Connected Generalized Inverse Limits and Intermediate Value Property

TL;DR

This paper explores conditions under which inverse limits of the interval [0,1] with set-valued functions are connected, introducing two generalizations of the Intermediate Value Property and analyzing their implications.

Contribution

It introduces two new generalizations of the Intermediate Value Property for set-valued functions and establishes their role in ensuring the connectedness of inverse limits.

Findings

Inverse limits are connected if bonding functions are surjective, have connected graphs, and satisfy the generalized Intermediate Value Property.

Examples show that dropping any condition can lead to disconnected inverse limits.

The paper compares different structures of set-valued functions with respect to the new properties.

Abstract

In this paper, we consider inverse limits of $[0,1]$ using upper semicontinuous set-valued functions. We introduce two generalizations of the Intermediate Value Property and prove that inverse limits with upper semicontinuous set-valued bonding functions are connected if the bonding functions are surjective, have connected graphs, and have either generalization of the Intermediate Value Property. Examples are given to demonstrate that if any of the conditions is dropped, the result does not hold in general. An example is given to show that an inverse limit may be connected even if the bonding functions do not have either Intermediate Value Property. Further, we compare the structures of set-valued functions with the two types of the Intermediate Value Property.