On the dimension of almost $n$-dimensional spaces
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- by M. Levin and E. D. Tymchatyn PDF
- Proc. Amer. Math. Soc. 127 (1999), 2793-2795 Request permission
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
- Mladen Bestvina, Characterizing $k$-dimensional universal Menger compacta, Mem. Amer. Math. Soc. 71 (1988), no. 380, vi+110. MR 920964, DOI 10.1090/memo/0380
- Beverly L. Brechner, On the dimensions of certain spaces of homeomorphisms, Trans. Amer. Math. Soc. 121 (1966), 516–548. MR 187208, DOI 10.1090/S0002-9947-1966-0187208-2
- Ryszard Engelking, Theory of dimensions finite and infinite, Sigma Series in Pure Mathematics, vol. 10, Heldermann Verlag, Lemgo, 1995. MR 1363947
- W. Hurewicz, Sur la dimension des produits Cartesiens, Ann. of Math., 36(1935), 194-197.
- Kazuhiro Kawamura, Lex G. Oversteegen, and E. D. Tymchatyn, On homogeneous totally disconnected $1$-dimensional spaces, Fund. Math. 150 (1996), no. 2, 97–112. MR 1391294, DOI 10.4064/fm-150-2-97-112
- Michael Levin and Roman Pol, A metric condition which implies dimension $\leq 1$, Proc. Amer. Math. Soc. 125 (1997), no. 1, 269–273. MR 1389528, DOI 10.1090/S0002-9939-97-03856-2
- T. Nishiura and E. D. Tymchatyn, Hereditarily locally connected spaces, Houston J. Math. 2 (1976), no. 4, 581–599. MR 436072
- V. Andriano and A. Bacciotti, A topological criterion for global asymptotic stability, Riv. Mat. Univ. Parma (5) 2 (1993), 313–318 (1994) (English, with Italian summary). MR 1276065
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
- MR Author ID: 175580
- Email: tymchatyn@math.usask.ca
- 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
- © Copyright 1999 American Mathematical Society
- 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
- MathSciNet review: 1600109