A dimension raising hereditary shape equivalence

Jan Dijkstra

Fundamenta Mathematicae (1996)

  • Volume: 149, Issue: 3, page 265-274
  • ISSN: 0016-2736

Abstract

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We construct a hereditary shape equivalence that raises transfinite inductive dimension from ω to ω+1. This shows that ind and Ind do not admit a geometric characterisation in the spirit of Alexandroff's Essential Mapping Theorem, answering a question asked by R. Pol.

How to cite

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Dijkstra, Jan. "A dimension raising hereditary shape equivalence." Fundamenta Mathematicae 149.3 (1996): 265-274. <http://eudml.org/doc/212123>.

@article{Dijkstra1996,
abstract = {We construct a hereditary shape equivalence that raises transfinite inductive dimension from ω to ω+1. This shows that ind and Ind do not admit a geometric characterisation in the spirit of Alexandroff's Essential Mapping Theorem, answering a question asked by R. Pol.},
author = {Dijkstra, Jan},
journal = {Fundamenta Mathematicae},
keywords = {hereditary shape equivalence; transfinite inductive dimension},
language = {eng},
number = {3},
pages = {265-274},
title = {A dimension raising hereditary shape equivalence},
url = {http://eudml.org/doc/212123},
volume = {149},
year = {1996},
}

TY - JOUR
AU - Dijkstra, Jan
TI - A dimension raising hereditary shape equivalence
JO - Fundamenta Mathematicae
PY - 1996
VL - 149
IS - 3
SP - 265
EP - 274
AB - We construct a hereditary shape equivalence that raises transfinite inductive dimension from ω to ω+1. This shows that ind and Ind do not admit a geometric characterisation in the spirit of Alexandroff's Essential Mapping Theorem, answering a question asked by R. Pol.
LA - eng
KW - hereditary shape equivalence; transfinite inductive dimension
UR - http://eudml.org/doc/212123
ER -

References

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  1. [1] P. S. Alexandroff, Dimensionstheorie. Ein Beitrag zur Geometrie der abgeschlossenen Mengen, Math. Ann. 106 (1932), 161-238. Zbl58.0624.01
  2. [2] F. D. Ancel, The role of countable dimensionality in the theory of cell-like relations, Trans. Amer. Math. Soc. 287 (1985), 1-40. Zbl0507.54017
  3. [3] F. D. Ancel, Proper hereditary shape equivalences preserve Property C, Topology Appl. 19 (1985), 71-74. Zbl0568.54017
  4. [4] P. Borst, Transfinite classifications of weakly infinite-dimensional spaces, Free University Press, Amsterdam, 1986. 
  5. [5] P. Borst and J. J. Dijkstra, Essential mappings and transfinite dimension, Fund. Math. 125 (1985), 41-45. Zbl0588.54033
  6. [6] J. J. Dijkstra, J. van Mill, and J. Mogilski, An AR-map whose range is more infinite-dimensional than its domain, Proc. Amer. Math. Soc. 114 (1992), 279-285. Zbl0765.54028
  7. [7] J. J. Dijkstra and J. Mogilski, A geometric approach to the dimension theory of infinite-dimensional spaces, in: Proc. 8th Ann. Workshop Geom. Topology, Univ. of Wisconsin-Milwaukee, 1991, 59-63. 
  8. [8] T. Dobrowolski and L. Rubin, The hyperspaces of infinite-dimensional compacta for covering and cohomological dimension are homeomorphic, Pacific J. Math. 164 (1994), 15-39. Zbl0801.54005
  9. [9] A. N. Dranišnikov [A. N. Dranishnikov], On a problem of P. S. Alexandrov, Mat. Sb. 135 (1988), 551-557 (in Russian). 
  10. [10] R. Engelking, Dimension Theory, North-Holland, Amsterdam, 1978. 
  11. [11] R. Geoghegan and R. R. Summerhill, Pseudo-boundaries and pseudo-interiors in euclidean spaces and topological manifolds, Trans. Amer. Math. Soc. 194 (1974), 141-165. Zbl0288.57001
  12. [12] W. E. Haver, Mappings between ANR's that are fine homotopy equivalences, Pacific J. Math. 58 (1975), 457-461. Zbl0311.55006
  13. [13] D. W. Henderson, A lower bound for transfinite dimension, Fund. Math. 64 (1968), 167-173. Zbl0167.51301
  14. [14] W. Hurewicz, Ueber unendlich-dimensionale Punktmengen, Proc. Akad. Amsterdam 31 (1928), 916-922. Zbl54.0620.05
  15. [15] G. Kozlowski, Images of ANR's, unpublished manuscript. 
  16. [16] J. van Mill and J. Mogilski, Property C and fine homotopy equivalences, Proc. Amer. Math. Soc. 90 (1984), 118-120. Zbl0548.57006
  17. [17] R. Pol, On a classification of weakly infinite-dimensional compacta, Topology Proc. 5 (1980), 231-242. Zbl0473.54024
  18. [18] R. Pol, On classification of weakly infinite-dimensional compacta, Fund. Math. 116 (1983), 169-188. Zbl0571.54030
  19. [19] Ju. M. Smirnov, On universal spaces for certain classes of infinite dimensional spaces, Amer. Math. Soc. Transl. Ser. 2, 21 (1962), 21-33. 

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