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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the dimension of certain totally disconnected spaces


Authors: Lex G. Oversteegen and E. D. Tymchatyn
Journal: Proc. Amer. Math. Soc. 122 (1994), 885-891
MSC: Primary 54F45; Secondary 54F15, 54G05, 54H20
MathSciNet review: 1273515
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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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1273515-1
PII: S 0002-9939(1994)1273515-1
Keywords: Totally disconnected, dimension, R-tree, Menger continua, homeomorphism groups, hereditarily locally connected
Article copyright: © Copyright 1994 American Mathematical Society