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

DOI:
https://doi.org/10.1090/S0002-9939-1992-1075946-X

MathSciNet review:
1075946

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Abstract: We construct an example of an AR-map , where is a strongly countable dimensional compact AR and is a countable dimensional AR which is not strongly countable dimensional. Using this map we find a shrinkable decomposition of the pre-Hilbert space 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.

**[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**and*, 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 MR**712860 (85d:57013)****[10]**G. Kozlowski,*Images of ANR's*, unpublished manuscript.**1.**ll.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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1992-1075946-X

Article copyright:
© Copyright 1992
American Mathematical Society