On the dimension of certain totally disconnected spaces

Oversteegen, Lex G.; Tymchatyn, E. D.

doi:10.1090/S0002-9939-1994-1273515-1

On the dimension of certain totally disconnected spaces
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by Lex G. Oversteegen and E. D. Tymchatyn PDF

Proc. Amer. Math. Soc. 122 (1994), 885-891 Request permission

Abstract:

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 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.

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