On the dimension of almost -dimensional spaces

Authors:
M. Levin and E. D. Tymchatyn

Journal:
Proc. Amer. Math. Soc. **127** (1999), 2793-2795

MSC (1991):
Primary 54F45, 54F25, 54F50

DOI:
https://doi.org/10.1090/S0002-9939-99-04846-7

Published electronically:
April 15, 1999

MathSciNet review:
1600109

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Abstract | References | Similar Articles | Additional Information

Abstract: Oversteegen and Tymchatyn proved that homeomorphism groups of positive dimensional Menger compacta are -dimensional by proving that almost -dimensional spaces are at most -dimensional. These homeomorphism groups are almost -dimensional and at least -dimensional by classical results of Brechner and Bestvina. In this note we prove that almost -dimensional spaces for are -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 -dimensional.

**1.**Mladen Bestvina, Characterizing -dimensional universal Menger compacta, Memoirs Amer. Math. Soc., 380(1988). MR**89g:54083****2.**Beverly Brechner, On the dimension of certain spaces of homeomorphisms, Trans. Amer. Math. Soc., 121(1966), 516-548. MR**32:4662****3.**R. Engelking, Theory of dimensions finite and infinite, Heldermann Verlag, Lemgo, 1995. MR**97j:54033****4.**W. Hurewicz, Sur la dimension des produits Cartesiens, Ann. of Math., 36(1935), 194-197.**5.**K. Kawamura, Lex G. Oversteegen and E. D. Tymchatyn, On homogeneous, totally disconnected, 1-dimensional spaces, Fund. Math., 150(1996), 97-112. MR**97d:54060****6.**Michael Levin and Roman Pol, A metric condition which implies dimension , Proc. Amer. Math. Soc., 125(1997), no. 1, 269-273. MR**97e:54033****7.**T. Nishiura and E. D. Tymchatyn, Hereditarily locally connected spaces, Houston J. Math., 2(1976), 581-599. MR**55:9023****8.**Lex G. Oversteegen and E. D. Tymchatyn, On the dimension of certain totally disconnected spaces, Proc. Amer. Math. Soc., 122(1994), 885-891. MR**95b:54050**

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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:
https://doi.org/10.1090/S0002-9939-99-04846-7

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