A Hereditarily Indecomposable Inverse Limit of Finite Path Graphs

Abstract

In this dissertation, we explore the properties and significance of inverse limit spaces where the factor spaces are path graphs. We define the graph topology for finite graphs and discuss the properties of a continuous map between graphs as well as properties of a traditional inverse limit of graphs. Most importantly, that a traditional inverse limit of finite path graphs is non-Hausdorff. We introduce a generalized inverse limit, where the first space is a metric arc and all other spaces are finite path graphs. By example, a technique is shown for constructing a generalized inverse limit, where the first space is a metric arc and the others are finite path graphs, that is homeomorphic to a traditional inverse limit of Hausdorff arcs.

Using crooked chains, we construct and analyze a non-Hausdorff hereditarily indecomposable continuum. This continuum has some interesting properties. These properties and the continuum's relationship with the Pseudo-arc is discussed. Ongoing work and open problems are stated.