Universal spaces in the theory of transfinite dimension, I

Wojciech Olszewski

Fundamenta Mathematicae (1994)

  • Volume: 144, Issue: 3, page 243-258
  • ISSN: 0016-2736

Abstract

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R. Pol has shown that for every countable ordinal α, there exists a universal space for separable metrizable spaces X with ind X = α . We prove that for every countable limit ordinal λ, there is no universal space for separable metrizable spaces X with Ind X = λ. This implies that there is no universal space for compact metrizable spaces X with Ind X = λ. We also prove that there is no universal space for compact metrizable spaces X with ind X = λ.

How to cite

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Olszewski, Wojciech. "Universal spaces in the theory of transfinite dimension, I." Fundamenta Mathematicae 144.3 (1994): 243-258. <http://eudml.org/doc/212027>.

@article{Olszewski1994,
abstract = {R. Pol has shown that for every countable ordinal α, there exists a universal space for separable metrizable spaces X with ind X = α . We prove that for every countable limit ordinal λ, there is no universal space for separable metrizable spaces X with Ind X = λ. This implies that there is no universal space for compact metrizable spaces X with Ind X = λ. We also prove that there is no universal space for compact metrizable spaces X with ind X = λ.},
author = {Olszewski, Wojciech},
journal = {Fundamenta Mathematicae},
keywords = {transfinite dimension; universal space; separable metric spaces},
language = {eng},
number = {3},
pages = {243-258},
title = {Universal spaces in the theory of transfinite dimension, I},
url = {http://eudml.org/doc/212027},
volume = {144},
year = {1994},
}

TY - JOUR
AU - Olszewski, Wojciech
TI - Universal spaces in the theory of transfinite dimension, I
JO - Fundamenta Mathematicae
PY - 1994
VL - 144
IS - 3
SP - 243
EP - 258
AB - R. Pol has shown that for every countable ordinal α, there exists a universal space for separable metrizable spaces X with ind X = α . We prove that for every countable limit ordinal λ, there is no universal space for separable metrizable spaces X with Ind X = λ. This implies that there is no universal space for compact metrizable spaces X with Ind X = λ. We also prove that there is no universal space for compact metrizable spaces X with ind X = λ.
LA - eng
KW - transfinite dimension; universal space; separable metric spaces
UR - http://eudml.org/doc/212027
ER -

References

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  1. [1] R. Engelking, Dimension Theory, PWN, Warszawa, 1978. 
  2. [2] R. Engelking, Transfinite dimension, in: Surveys in General Topology, G. M. Reed (ed.), Academic Press, 1980, 131-161. 
  3. [3] R. Engelking, General Topology, Heldermann, Berlin, 1989. 
  4. [4] W. Hurewicz, Ueber unendlich-dimensionale Punktmengen, Proc. Akad. Amsterdam 31 (1928), 167-173. 
  5. [5] M. Landau, Strong transfinite ordinal dimension, Bull. Amer. Math. Soc. 21 (1969), 591-596. Zbl0175.19903
  6. [6] B. T. Levšenko [B. T. Levshenko], Spaces of transfinite dimensionality, Mat. Sb. 67 (1965), 255-266 (in Russian); English transl.: Amer. Math. Soc. Transl. Ser. 2 73 (1968), 135-148. 
  7. [7] L. A. Luxemburg, On infinite-dimensional spaces with transfinite dimension, Dokl. Akad. Nauk SSSR 199 (1971), 1243-1246 (in Russian); English transl.: Soviet Math. Dokl. 12 (1971), 1272-1276. Zbl0235.54036
  8. [8] L. A. Luxemburg, On transfinite inductive dimensions, Dokl. Akad. Nauk SSSR 209 (1973), 295-298 (in Russian); English transl.: Soviet Math. Dokl. 14 (1973), 388-393. Zbl0283.54019
  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] L. A. Luxemburg, On compactifications of metric spaces with transfinite dimensions, Pacific J. Math. 101 (1982), 399-450. Zbl0451.54030
  11. [11] L. A. Luxemburg, On universal infinite-dimensional spaces, Fund. Math. 122 (1984), 129-147. Zbl0571.54029
  12. [12] A. R. Pears, A note on transfinite dimension, ibid. 71 (1971), 215-221. 
  13. [13] R. Pol, On classification of weakly infinite-dimensional compacta, ibid. 116 (1983), 169-188. Zbl0571.54030
  14. [14] R. Pol, Countable-dimensional universal sets, Trans. Amer. Math. Soc. 297 (1986), 255-268. Zbl0636.54032
  15. [15] R. Pol, Questions in dimension theory, in: Open Problems in Topology, J. van Mill and G. M. Reed (eds.), North-Holland, 1990, 279-291. 
  16. [16] 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. Ser. 2 21 (1962), 21-33. 
  17. [17] G. H. Toulmin, Shuffling ordinals and transfinite dimension, Proc. London Math. Soc. 4 (1954), 177-195. Zbl0055.41406

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