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A counterexample concerning line-free groups and complete Erdos space
Authors:
Jan J. Dijkstra and Jan van Mill
Journal:
Proc. Amer. Math. Soc. 134 (2006), 2281-2283
MSC (2000):
Primary 46B25, 54F45
Posted:
February 2, 2006
MathSciNet review:
2213700
Full-text PDF Free Access
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Additional Information
Abstract: We present a weakly closed, one-dimensional, line-free subgroup of the separable Banach space  that is not homeomorphic to complete Erdos space. The existence of this example disproves a conjecture of Dobrowolski, Grabowski, and Kawamura.
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D. Ancel, Tadeusz
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Grabowski, Closed subgroups in Banach spaces, Studia Math.
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(95d:46009)
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Amer. Math. Soc. 10
(2004), 29–38 (electronic). MR 2048429
(2005c:57025), http://dx.doi.org/10.1090/S1079-6762-04-00127-1
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J. J. Dijkstra and J. van Mill, Erdos space and homeomorphism groups of manifolds, preprint.
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J. J. Dijkstra and J. van Mill, Characterizing complete Erdos space, preprint.
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Jan
J. Dijkstra, Jan
Van Mill, and Juris
Steprāns, Complete Erdős space is unstable,
Math. Proc. Cambridge Philos. Soc. 137 (2004), no. 2,
465–473. MR 2092071
(2005g:54076), http://dx.doi.org/10.1017/S0305004104007996
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1391294 (97d:54060)
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- 1.
- M. Abry and J. J. Dijkstra, On topological Kadec norms, Math. Ann. 332 (2005), 759-765. MR 2179775
- 2.
- F. D. Ancel, T. Dobrowolski, and J. Grabowski, Closed subgroups in Banach spaces, Studia Math. 109 (1994), 277-290. MR 1274013 (95d:46009)
- 3.
- J. J. Dijkstra and J. van Mill, Homeomorphism groups of manifolds and Erdos space, Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 29-38. MR 2048429 (2005c:57025)
- 4.
- J. J. Dijkstra and J. van Mill, Erdos space and homeomorphism groups of manifolds, preprint.
- 5.
- J. J. Dijkstra and J. van Mill, Characterizing complete Erdos space, preprint.
- 6.
- J. J. Dijkstra, J. van Mill and J. Steprans, Complete Erdos space is unstable, Math. Proc. Cambridge Philos. Soc. 137 (2004), 465-473. MR 2092071
- 7.
- T. Dobrowolski, J. Grabowski, and K. Kawamura, Topological type of weakly closed subgroups in Banach spaces, Studia Math. 118 (1996), 49-62. MR 1373624 (97d:46013)
- 8.
- P. Erdos, The dimension of the rational points in Hilbert space, Ann. of Math. 41 (1940), 734-736. MR 0003191 (2:178a)
- 9.
- K. Kawamura, L. G. Oversteegen, and E. D. Tymchatyn, On homogeneous totally disconnected
 -dimensional spaces, Fund. Math. 150 (1996), 97-112. MR 1391294 (97d:54060)
- 10.
- M. Levin and R. Pol, A metric condition which implies dimension
 , Proc. Amer. Math. Soc. 125 (1997), 269-273. MR 1389528 (97e:54033)
- 11.
- L. G. Oversteegen and E. D. Tymchatyn, On the dimension of certain totally disconnected spaces, Proc. Amer. Math. Soc. 122 (1994), 885-891. MR 1273515 (95b:54040)
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Additional Information
Jan J. Dijkstra
Affiliation:
Faculteit der Exacte Wetenschappen/Afdeling Wiskunde, Vrije Universiteit, De Boelelaan 1081$^{a}$, 1081 HV Amsterdam, The Netherlands
Email:
dijkstra@cs.vu.nl
Jan van Mill
Affiliation:
Faculteit der Exacte Wetenschappen/Afdeling Wiskunde, Vrije Universiteit, De Boelelaan 1081$^{a}$, 1081 HV Amsterdam, The Netherlands
Email:
vanmill@cs.vu.nl
DOI:
http://dx.doi.org/10.1090/S0002-9939-06-08232-3
PII:
S 0002-9939(06)08232-3
Keywords:
Banach space,
line-free group,
weakly closed,
complete Erd\H os space,
almost zero-dimensional
Received by editor(s):
December 8, 2004
Received by editor(s) in revised form:
February 25, 2005
Posted:
February 2, 2006
Communicated by:
N. Tomczak-Jaegermann
Article copyright:
© Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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