On the connectivity of graph Lipscomb’s spaces

Abstract A central role in topological dimension theory is played by Lipscomb’s space $$J_{A}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>J</mml:mi> <mml:mi>A</mml:mi> </mml:msub> </mml:math> since it is a universal space for metric spaces of weight $$|A|\ge \aleph _{0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>A</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:mo>≥</mml:mo> <mml:msub> <mml:mi>ℵ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:math> . 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 Miculescu and Mihail in Aequat Math, 96:1141–1157, 2022). It is denoted by $$J_{A}^{\mathcal {G}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>J</mml:mi> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:mi>G</mml:mi> </mml:msubsup> </mml:math> and is called the graph Lipscomb’s space associated with the graph $$\mathcal {G}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> 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 $$\mathcal {G}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> for which $$J_{A}^{\mathcal {G}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>J</mml:mi> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:mi>G</mml:mi> </mml:msubsup> </mml:math> is connected. In the particular case when A is finite, some supplementary characterizations are presented.