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An AR-map whose range is more infinite-dimensional than its domain

Authors: J. J. Dijkstra, J. van Mill and J. Mogilski
Journal: Proc. Amer. Math. Soc. 114 (1992), 279-285
MSC: Primary 54C55; Secondary 54G20, 57N20
MathSciNet review: 1075946
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Abstract: We construct an example of an AR-map $ f:X \to Y$, where $ X$ is a strongly countable dimensional compact AR and $ Y$ is a countable dimensional AR which is not strongly countable dimensional. Using this map we find a shrinkable decomposition of the pre-Hilbert space $ l_f^2$ whose quotient map does not stabilize to a near homeomorphism. We also present a partial result concerning the question whether cell-like maps preserve countable dimensionality.

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  • [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] C. Bessaga and A. Pelczyński, Selected topics in infinite-dimensional topology, PWN, Warsaw, 1975.
  • [3] K. Borsuk, Shape theory, PWN, Warsaw, 1975.
  • [4] A. N. Dranišnikov, On a problem of P. S. Alexandrov, Mat. Sb. 135 (1988), 551-557.
  • [5] J. Dydak and J. Segal, Shape theory: An introduction, Lectures Notes in Math., vol. 688, Springer-Verlag, Berlin, 1978. MR 520227 (80h:54020)
  • [6] R. Engelking and E. Pol, Countable dimensional spaces. A survey, Dissertationes Math. 216 (1983), 1-45. MR 722011 (85f:54075)
  • [7] W. E. Haver, Mappings between ANR's that are fine homotopy equivalence, Pacific J. Math. 58 (1975), 457-461. MR 0385865 (52:6724)
  • [8] D. W. Henderson, A lower bound for transfinite dimension, Fund. Math. 64 (1968), 167-173. MR 0243496 (39:4817)
  • [9] J. P. Henderson and J. J. Walsh, Examples of cell-like decompositions if the infinite dimensional manifolds $ \sigma $ and $ \Sigma $, Topology Appl. 16 (1983), 143-154. MR 712860 (85d:57013)
  • [10] G. Kozlowski, Images of ANR's, unpublished manuscript.
  • 1. ll. J. van Mill, Infinite-dimensional topology, prerequisites and introduction, North-Holland, Amsterdam, 1989. MR 977744 (90a:57025)
  • [12] R. Pol, On classification of weakly infinite dimensional compacta, Fund. Math. 116 (1983), 169-188. MR 716218 (85g:54029)
  • [13] J. E. West, Open problems in infinite-dimensional topology, Open Problems in Topology (J. van Mill and G. M. Reed, eds.), North-Holland, Amsterdam, 1990, pp. 523-597. MR 1078666
  • [14] J. J. Dijkstra and J. Mogilski, in preparation.

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Article copyright: © Copyright 1992 American Mathematical Society

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