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)

 
 

 

Local contractibility, cell-like maps, and dimension


Author: Jan van Mill
Journal: Proc. Amer. Math. Soc. 98 (1986), 534-536
MSC: Primary 55M10; Secondary 54C35
DOI: https://doi.org/10.1090/S0002-9939-1986-0857957-7
MathSciNet review: 857957
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the existence of cell-like maps $ f:$: $ {I^n} \to X$ such that no nonempty open subset of $ X$ is contractible in $ X$. From the Taylor Example, it is easy to construct such a map for $ n = \infty $. We show that there exists such a map for some finite $ n$ if (and only if) there exists a dimension raising cell-like map of a compactum.


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

  • [1] F. D. Ancel, The role of countable dimensionality in the theory of cell-like relations, Trans. Amer. Math. Soc. 287 (1985), 1-40. MR 766204 (86b:54012)
  • [2] K. Borsuk, Theory of retracts, PWN, Warsaw, 1967. MR 0216473 (35:7306)
  • [3] R. J. Daverman and J. J. Walsh, Example of cell-like maps that are not shape equivalences, Michigan Math. J. 30 (1983), 17-30. MR 694925 (85g:57008)
  • [4] R. Engelking, Dimension theory, PWN, Warsaw, 1978. MR 0482697 (58:2753b)
  • [5] J. Keesling, A non-movable trivial-shape decomposition of the Hilbert cube, Bull. Acad. Polon. Sci. 23 (1975), 997-998. MR 0391100 (52:11922)
  • [6] G. Kozlowski, Images of ANR's, unpublished manuscript.
  • [7] J. L. Taylor, A counterexample in shape theory, Bull. Amer. Math. Soc. 81 (1975), 629-632. MR 0375328 (51:11523)
  • [8] J. J. Walsh, Dimension, cohomological dimension, and cell-like mappings, Shape Theory and Geometric Topology, Lecture Notes in Math., vol. 870, Springer-Verlag, Berlin and New York, 1981, pp. 105-118. MR 643526 (83a:57021)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 55M10, 54C35

Retrieve articles in all journals with MSC: 55M10, 54C35


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1986-0857957-7
Keywords: Cell-like map, dimension, local contractibility
Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society