Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. Mladen Bestvina, Characterizing $k$-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 $\leq 1$, 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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 54F45, 54F25, 54F50

Retrieve articles in all journals with MSC (1991): 54F45, 54F25, 54F50

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

E. D. Tymchatyn
Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Canada S7N 0W0

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

American Mathematical Society