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.
@article{bwmeta1.element.bwnjournal-article-fmv149i3p265bwm,
author = {Jan Dijkstra},
title = {A dimension raising hereditary shape equivalence},
journal = {Fundamenta Mathematicae},
volume = {149},
year = {1996},
pages = {265-274},
zbl = {0855.54046},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv149i3p265bwm}
}
Dijkstra, Jan. A dimension raising hereditary shape equivalence. Fundamenta Mathematicae, Tome 149 (1996) pp. 265-274. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv149i3p265bwm/
[00000] [1] P. S. Alexandroff, Dimensionstheorie. Ein Beitrag zur Geometrie der abgeschlossenen Mengen, Math. Ann. 106 (1932), 161-238. | Zbl 58.0624.01
[00001] [2] F. D. Ancel, The role of countable dimensionality in the theory of cell-like relations, Trans. Amer. Math. Soc. 287 (1985), 1-40. | Zbl 0507.54017
[00002] [3] F. D. Ancel, Proper hereditary shape equivalences preserve Property C, Topology Appl. 19 (1985), 71-74. | Zbl 0568.54017
[00003] [4] P. Borst, Transfinite classifications of weakly infinite-dimensional spaces, Free University Press, Amsterdam, 1986.
[00004] [5] P. Borst and J. J. Dijkstra, Essential mappings and transfinite dimension, Fund. Math. 125 (1985), 41-45. | Zbl 0588.54033
[00005] [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. | Zbl 0765.54028
[00006] [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.
[00007] [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. | Zbl 0801.54005
[00008] [9] A. N. Dranišnikov [A. N. Dranishnikov], On a problem of P. S. Alexandrov, Mat. Sb. 135 (1988), 551-557 (in Russian).
[00009] [10] R. Engelking, Dimension Theory, North-Holland, Amsterdam, 1978.
[00010] [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. | Zbl 0288.57001
[00011] [12] W. E. Haver, Mappings between ANR's that are fine homotopy equivalences, Pacific J. Math. 58 (1975), 457-461. | Zbl 0311.55006
[00012] [13] D. W. Henderson, A lower bound for transfinite dimension, Fund. Math. 64 (1968), 167-173. | Zbl 0167.51301
[00013] [14] W. Hurewicz, Ueber unendlich-dimensionale Punktmengen, Proc. Akad. Amsterdam 31 (1928), 916-922. | Zbl 54.0620.05
[00014] [15] G. Kozlowski, Images of ANR's, unpublished manuscript.
[00015] [16] J. van Mill and J. Mogilski, Property C and fine homotopy equivalences, Proc. Amer. Math. Soc. 90 (1984), 118-120. | Zbl 0548.57006
[00016] [17] R. Pol, On a classification of weakly infinite-dimensional compacta, Topology Proc. 5 (1980), 231-242. | Zbl 0473.54024
[00017] [18] R. Pol, On classification of weakly infinite-dimensional compacta, Fund. Math. 116 (1983), 169-188. | Zbl 0571.54030
[00018] [19] Ju. M. Smirnov, On universal spaces for certain classes of infinite dimensional spaces, Amer. Math. Soc. Transl. Ser. 2, 21 (1962), 21-33.