On the connectivity of graph Lipscomb's space

TL;DR

This paper characterizes the conditions under which graph Lipscomb's spaces, a generalization of Lipscomb's space using graphs, are connected, revealing links between topological dimension theory and fractal set theory.

Contribution

It provides a characterization of graphs that produce connected graph Lipscomb's spaces, extending understanding of their topological properties and fractal nature.

Findings

Identifies conditions for connectivity of graph Lipscomb's spaces.

Shows that graph Lipscomb's space is a generalized Hutchinson-Barnsley fractal.

Provides additional characterizations when the set A is finite.

Abstract

A central role in topological dimension theory is played by Lipscomb's space $J_{A}$ since it is a universal space for metric spaces of weight $|A|\geq \aleph _{0}$. On the one hand, Lipscomb's space is the attractor of a possibly infinite iterated function system, i.e. it is a generalized Hutchinson-Barnsley fractal. As, on the other hand, some classical fractal sets are universal spaces, one can conclude that there exists a strong connection between topological dimension theory and fractal set theory. A generalization of Lipscomb's space, using graphs, has been recently introduced (see R. Miculescu, A. Mihail, Graph Lipscomb's space is a generalized Hutchinson-Barnsley fractal, Aequat. Math., \textbf{96} (2022), 1141-1157). It is denoted by $J_{A}^{\G}$ and it is called graph Lipscomb's space associated with the graph $\G$ on the set $A$. It turns out that it is a topological copy of a generalized Hutchinson-Barnsley fractal. This paper provides a characterization of those graphs $\G$ for which $J_{A}^{\G}$ is connected. In the particular case when $A$ is finite, some supplementary characterizations are presented.