ABOUT GEHMAN DENDRITE G3 AS GENERALIZED INVERSE LIMIT SPACE

Abstract

This paper introduces a new uncountable family of upper semi-continuous set-valued functions whose generalized inverse limits are homeomorphic to the Gehman dendrite  , a key object in continuum theory characterized by ramification points of order three and a Cantor set of endpoints. While previous work by Farhan and Mena established one such family, our construction extends their results, providing a distinct class of bonding functions defined on the unit interval using specific parameters and graph structures. Our main results prove that for any function in these families, the inverse limit space is , which we establish by showing its ramification points are of order three and its endpoints form a Cantor set. We provide illustrative examples and counterexamples that demonstrate the necessity of our conditions. This work broadens the catalog of set-valued dynamical systems that produce classic topological structures and highlights the nuanced relationship between bonding function graphs and their inverse limits.