Cantor manifolds in the theory of transfinite dimension

Wojciech Olszewski

Fundamenta Mathematicae (1994)

  • Volume: 145, Issue: 1, page 39-64
  • ISSN: 0016-2736

Abstract

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For every countable non-limit ordinal α we construct an α-dimensional Cantor ind-manifold, i.e., a compact metrizable space Z α such that i n d Z α = α , and no closed subset L of Z α with ind L less than the predecessor of α is a partition in Z α . An α-dimensional Cantor Ind-manifold can be constructed similarly.

How to cite

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Olszewski, Wojciech. "Cantor manifolds in the theory of transfinite dimension." Fundamenta Mathematicae 145.1 (1994): 39-64. <http://eudml.org/doc/212033>.

@article{Olszewski1994,
abstract = {For every countable non-limit ordinal α we construct an α-dimensional Cantor ind-manifold, i.e., a compact metrizable space $Z_α$ such that $ind Z_α = α$, and no closed subset L of $Z_α$ with ind L less than the predecessor of α is a partition in $Z_α$. An α-dimensional Cantor Ind-manifold can be constructed similarly.},
author = {Olszewski, Wojciech},
journal = {Fundamenta Mathematicae},
keywords = {small inductive dimension; large inductive dimension; transfinite dimension; Cantor manifold},
language = {eng},
number = {1},
pages = {39-64},
title = {Cantor manifolds in the theory of transfinite dimension},
url = {http://eudml.org/doc/212033},
volume = {145},
year = {1994},
}

TY - JOUR
AU - Olszewski, Wojciech
TI - Cantor manifolds in the theory of transfinite dimension
JO - Fundamenta Mathematicae
PY - 1994
VL - 145
IS - 1
SP - 39
EP - 64
AB - For every countable non-limit ordinal α we construct an α-dimensional Cantor ind-manifold, i.e., a compact metrizable space $Z_α$ such that $ind Z_α = α$, and no closed subset L of $Z_α$ with ind L less than the predecessor of α is a partition in $Z_α$. An α-dimensional Cantor Ind-manifold can be constructed similarly.
LA - eng
KW - small inductive dimension; large inductive dimension; transfinite dimension; Cantor manifold
UR - http://eudml.org/doc/212033
ER -

References

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  1. [1] V. A. Chatyrko, Counterparts of Cantor manifolds for transfinite dimensions, Mat. Zametki 42 (1987), 115-119 (in Russian). Zbl0642.54028
  2. [2] R. Engelking, Dimension Theory, PWN, Warszawa, 1978. 
  3. [3] R. Engelking, Transfinite dimension, in: Surveys in General Topology, G. M. Reed (ed.), Academic Press, 1980, 131-161. 
  4. [4] R. Engelking, General Topology, Heldermann, Berlin, 1989. 
  5. [5] D. W. Henderson, A lower bound for transfinite dimension, Fund. Math. 63 (1968), 167-173. Zbl0167.51301
  6. [6] W. Hurewicz, Ueber unendlich-dimensionale Punktmengen, Proc. Akad. Amsterdam 31 (1928), 916-922. Zbl54.0620.05
  7. [7] M. Landau, Strong transfinite ordinal dimension, Bull. Amer. Math. Soc. 21 (1969), 591-596. Zbl0175.19903
  8. [8] B. T. Levšenko [B. T. Levshenko], Spaces of transfinite dimensionality, Mat. Sb. 67 (1965), 255-266 (in Russian); English transl.: Amer. Math. Soc. Transl. (2) 72 (1968), 135-148. 
  9. [9] L. A. Luxemburg, On compact spaces with non-coinciding transfinite dimensions, Dokl. Akad. Nauk SSSR 212 (1973), 1297-1300 (in Russian); English transl.: Soviet Math. Dokl. 14 (1973), 1593-1597. 
  10. [10] W. Olszewski, Universal spaces in the theory of transfinite dimension, I, Fund. Math. 144 (1994), 243-258. Zbl0812.54041
  11. [11] A. R. Pears, A note on transfinite dimension, ibid. 71 (1971), 215-221. 
  12. [12] Yu. M. Smirnov, On universal spaces for certain classes of infinite dimensional spaces, Izv. Akad. Nauk SSSR Ser. Mat. 23 (1959), 185-196 (in Russian); English transl.: Amer. Math. Soc. Transl. (2) 21 (1962), 21-33. 
  13. [13] G. H. Toulmin, Shuffling ordinals and transfinite dimension, Proc. London Math. Soc. 4 (1954), 177-195. Zbl0055.41406

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