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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 almost $n$-dimensional spaces


Authors: M. Levin and E. D. Tymchatyn
Journal: Proc. Amer. Math. Soc. 127 (1999), 2793-2795
MSC (1991): Primary 54F45, 54F25, 54F50
Published electronically: April 15, 1999
MathSciNet review: 1600109
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Abstract: Oversteegen and Tymchatyn proved that homeomorphism groups of positive dimensional Menger compacta are $1$-dimensional by proving that almost $0$-dimensional spaces are at most $1$-dimensional. These homeomorphism groups are almost $0$-dimensional and at least $1$-dimensional by classical results of Brechner and Bestvina. In this note we prove that almost $n$-dimensional spaces for $n \geq 1$ are $n$-dimensional. As a corollary we answer in the affirmative an old question of R. Duda by proving that every hereditarily locally connected, non-degenerate, separable, metric space is $1$-dimensional.


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Additional Information

M. Levin
Affiliation: Department of Mathematics, Tulane University, New Orleans, Louisiana 70118-5698
Address at time of publication: Institute of Mathematics, Tsukuba University, Tsukuba, Ibaraki 305, Japan
Email: mlevin@mozart.math.tulane.edu, mlevin@math.tsukuba.ac.jp

E. D. Tymchatyn
Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Canada S7N 0W0
Email: tymchatyn@math.usask.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04846-7
PII: S 0002-9939(99)04846-7
Keywords: Almost $0$-dimensional spaces, $L$-embeddings, hereditarily locally connected spaces
Received by editor(s): February 13, 1997
Received by editor(s) in revised form: November 20, 1997
Published electronically: April 15, 1999
Additional Notes: The authors were supported in part by NSERC grant OGP0005616.
Communicated by: Alan Dow
Article copyright: © Copyright 1999 American Mathematical Society