On homogeneous totally disconnected 1-dimensional spaces. (English) Zbl 0861.54028

A separable metrizable space \(X\) is almost 0-dimensional if \(X\) has a basis \({\mathcal B}\) such that \(X \smallsetminus \overline B\) is a union of clopen sets for each \(B\in {\mathcal B}\). It is known that a topologically complete almost 0-dimensional space has dimension at most 1. One-dimensional examples include “complete Erdös space” (the set of irrational points in Hilbert space), the set of endpoints \(E(L)\) of the Lelek fan, and the set of endpoints of the Julia set of the exponential map. In this paper, the authors provide a characterization of \(E(L)\) in terms of a metric and a sequence of disjoint clopen coverings of \(X\). The result is applied to show (among other things) that the three examples mentioned above are all homeomorphic.


54F65 Topological characterizations of particular spaces
54H20 Topological dynamics (MSC2010)
54E35 Metric spaces, metrizability
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