Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On Lipschitz homogeneity of the Hilbert cube


Author: Aarno Hohti
Journal: Trans. Amer. Math. Soc. 291 (1985), 75-86
MSC: Primary 54E40; Secondary 57N20
DOI: https://doi.org/10.1090/S0002-9947-1985-0797046-7
MathSciNet review: 797046
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The main contribution of this paper is to prove the conjecture of [] that the Hilbert cube $ Q$ is Lipschitz homogeneous for any metric $ {d_s}$, where $ s$ is a decreasing sequence of positive real numbers $ {s_k}$ converging to zero, $ {d_s}(x,y) = \sup \{ {s_k}\vert{x_k} - {y_k}\vert:k \in N\} $, and $ R(s) = \sup \{ {s_k}/{s_{k + 1}}:k \in N\} < \infty $. In addition to other results, we shall show that for every Lipschitz homogeneous compact metric space $ X$ there is a constant $ \lambda < \infty $ such that $ X$ is homogeneous with respect to Lipschitz homeomorphisms whose Lipschitz constants do not exceed $ \lambda $. Finally, we prove that the hyperspace $ {2^I}$ of all nonempty closed subsets of the unit interval is not Lipschitz homogeneous with respect to the Hausdorff metric.


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

  • [Ch] T. A. Chapman, Lectures on Hilbert cube manifolds, CBMS Regional Conf. Ser. in Math., No. 28, Amer. Math. Soc., Providence, R.I., 1976. MR 0423357 (54:11336)
  • [CS] D. W. Curtis and R. M. Schori, Hyperspaces of Peano continua are Hilbert cubes, Fund. Math. 101 (1978), 19-38. MR 512241 (80k:54010)
  • [DW] D. van Dantzig and B. L. van der Waerden, Ueber metrisch homogene Räume, Abh. Math. Sem. Univ. Hamburg 6 (1928), 367-376.
  • [HJ] A. Hohti and H. Junnila, A note on homogeneous metrizable spaces, manuscript.
  • [Ke] O. H. Keller, Die Homoiomorphie der kompakten konvexen Mengen in Hilbertschen Raum, Math. Ann. 105 (1931), 748-758. MR 1512740
  • [SW] R. M. Schori and J. E. West, The hyperspace of the closed unit interval is a Hilbert cube, Trans. Amer. Math. Soc. 213 (1975), 217-235. MR 0390993 (52:11815)
  • [Vä] J. Väisälä, Lipschitz homeomorphisms of the Hilbert cube, Topology Appl. 11 (1980), 103-110. MR 550877 (81d:54024)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 54E40, 57N20

Retrieve articles in all journals with MSC: 54E40, 57N20


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0797046-7
Keywords: Hilbert cube, homogeneous, Lipschitz homeomorphism, $ Q$-manifold, hyperspace
Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society