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On the dimension of certain totally disconnected spaces. (English) Zbl 0817.54028

Summary: It is well known that there exist separable, metrizable, totally disconnected spaces of all dimensions. In this note we introduce the notion of an almost 0-dimensional space and prove that every such space is a totally disconnected subspace of an \(\mathbb{R}\)-tree and, hence, at most 1-dimensional. As applications we prove that the spaces of homeomorphisms of the universal Menger continua are 1-dimensional and that hereditarily locally connected spaces have dimension at most two.

MSC:

54H20 Topological dynamics (MSC2010)
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
54G05 Extremally disconnected spaces, \(F\)-spaces, etc.
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