## 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.

### MSC:

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

### Keywords:

totally disconnected; homogeneous; complete
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