Abstract
Being a universal space for weight \(\left| A\right| \ge \aleph _{0}\) metric spaces Lipscomb’s space \(J_{A}\) has a central role in topological dimension theory. There exists a strong connection between topological dimension theory and fractal set theory since on the one hand, some classical fractals play the role of universal spaces and on the other hand the universal space \(J_{A}\) is a generalized Hutchinson–Barnsley fractal (i.e. the attractor of a possibly infinite iterated function system). In this paper we introduce a generalization of \(J_{A}\), namely the concept of graph Lipscomb’s space \(J_{A}^{{\mathcal {G}}}\) associated with a graph \({\mathcal {G}}\) on the set A, and we prove that its imbedded version in \(l^{2}(A^{\prime })\), where \(A^{\prime }=A\setminus \{z\}\) , z being a fixed element of the set A having at least two elements, is a generalized Hutchinson–Barnsley fractal.
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The authors are very grateful to the reviewers whose extremely generous and valuable remarks and comments brought substantial improvements to the paper.
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Miculescu, R., Mihail, A. Graph Lipscomb’s space is a generalized Hutchinson–Barnsley fractal. Aequat. Math. 96, 1141–1157 (2022). https://doi.org/10.1007/s00010-022-00918-x
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DOI: https://doi.org/10.1007/s00010-022-00918-x