Countable-dimensional universal sets

Pol, Roman

doi:10.1090/S0002-9947-1986-0849478-7

Countable-dimensional universal sets

Author: Roman Pol
Journal: Trans. Amer. Math. Soc. 297 (1986), 255-268
MSC: Primary 54F45; Secondary 54F65
DOI: https://doi.org/10.1090/S0002-9947-1986-0849478-7
MathSciNet review: 849478
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The main results of this paper are a construction of a countable union of zero dimensional sets in the Hilbert cube whose complement does not contain any subset of finite dimension $n \geqslant 1$ (Theorem 2.1, Corollary 2.3) and a construction of universal sets for the transfinite extension of the Menger-Urysohn inductive dimension (Theorem 2.2, Corollary 2.4).

[Bo] N. Bourbaki, Topologie générale, Chapitre 1, Hermann, Paris, 1948.
[En 1] Ryszard Engelking, Teoria wymiaru, Państwowe Wydawnictwo Naukowe, Warsaw, 1977 (Polish). Biblioteka Matematyczna, Tom 51. [Mathematics Library, Vol. 51]. MR 0482696
[En 2] R. Engelking, Transfinite dimension, Surveys in general topology, Academic Press, New York-London-Toronto, Ont., 1980, pp. 131–161. MR 564101
[G-S] Dennis J. Garity and Richard M. Schori, Infinite-dimensional dimension theory, Proceedings of the 1985 topology conference (Tallahassee, Fla., 1985), 1985, pp. 59–74. MR 851201
[Hu 1] W. Hurewicz, Ueber unendlich-dimensionale Punktmengen, Proc. Akad. Amsterdam 31 (1928), 916-922.
[H-W] Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, v. 4, Princeton University Press, Princeton, N. J., 1941. MR 0006493
[J-R] J. E. Jayne and C. A. Rogers, Borel isomorphisms at the first level. I, Mathematika 26 (1979), no. 1, 125–156. MR 557137, https://doi.org/10.1112/S0025579300009712
[Kr] J. Krasinkiewicz, Essential mappings onto products of manifolds, Geometric and algebraic topology, Banach Center Publ., vol. 18, PWN, Warsaw, 1986, pp. 377–406. MR 925878
[Kn] Bronisław Knaster, Sur les coupures biconnexes des espaces euclidiens de dimension 𝑛>1 arbitraire, Rec. Math. [Mat. Sbornik] N.S. 19(61) (1946), 9–18 (Russian, with French summary). MR 0017520
[Ku] K. Kuratowski, Topology. Vol. I, New edition, revised and augmented. Translated from the French by J. Jaworowski, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. MR 0217751
[Le] A. Lelek, Dimension inequalities for unions and mappings of separable metric spaces, Colloq. Math. 23 (1971), 69–91. MR 322829, https://doi.org/10.4064/cm-23-1-69-91
[Lu 1] Leonid Luxemburg, On compactifications of metric spaces with transfinite dimensions, Pacific J. Math. 101 (1982), no. 2, 399–450. MR 675409
[Lu 2] Leonid A. Luxemburg, On universal infinite-dimensional spaces, Fund. Math. 122 (1984), no. 2, 129–147. MR 753020, https://doi.org/10.4064/fm-122-2-129-147
[Ma] S. Mazurkiewicz, Sur les problèmes $\kappa$ et $\lambda$ de Urysohn, Fund. Math. 10 (1926), 311-319.
[Mo] Yiannis N. Moschovakis, Descriptive set theory, Studies in Logic and the Foundations of Mathematics, vol. 100, North-Holland Publishing Co., Amsterdam-New York, 1980. MR 561709
[Na 1] J. Nagata, On the countable sum of zero-dimensional metric spaces, Fund. Math. 48 (1959/60), 1–14. MR 114203, https://doi.org/10.4064/fm-48-1-1-14
[Na 2] Jun-iti Nagata, Modern dimension theory, Revised edition, Sigma Series in Pure Mathematics, vol. 2, Heldermann Verlag, Berlin, 1983. MR 715431
[Po 1] Roman Pol, A weakly infinite-dimensional compactum which is not countable-dimensional, Proc. Amer. Math. Soc. 82 (1981), no. 4, 634–636. MR 614892, https://doi.org/10.1090/S0002-9939-1981-0614892-2
[Po 2] Roman Pol, On classification of weakly infinite-dimensional compacta, Fund. Math. 116 (1983), no. 3, 169–188. MR 716218, https://doi.org/10.4064/fm-116-3-169-188
[Ru 1] Leonard R. Rubin, Hereditarily strongly infinite-dimensional spaces, Michigan Math. J. 27 (1980), no. 1, 65–73. MR 555838
[Ru 2] Leonard R. Rubin, Noncompact hereditarily strongly infinite-dimensional spaces, Proc. Amer. Math. Soc. 79 (1980), no. 1, 153–154. MR 560602, https://doi.org/10.1090/S0002-9939-1980-0560602-6
[Ru 3] -, Construction of hereditarily strongly infinite dimensional spaces, preprint, December 1984.
[R-S-W] Leonard R. Rubin, R. M. Schori, and John J. Walsh, New dimension-theory techniques for constructing infinite-dimensional examples, General Topology Appl. 10 (1979), no. 1, 93–102. MR 519716, https://doi.org/10.1016/0016-660x(79)90031-x
[S-W] R. M. Schori and John J. Walsh, Examples of hereditarily strongly infinite-dimensional compacta, Topology Proc. 3 (1978), no. 2, 495–506 (1979). MR 540508
[Tu] L. Tumarkin, Concerning infinite-dimensional spaces, General Topology and its Relations to Modern Analysis and Algebra (Proc. Sympos., Prague, 1961) Academic Press, New York; Publ. House Czech. Acad. Sci., Prague, 1962, pp. 352–353. MR 0150727
[Ur] P. Urysohn, Memoire sur les multiplicités cantoriennes, Fund. Math. 7 (1925), 30-137.
[Wa] John J. Walsh, Infinite-dimensional compacta containing no 𝑛-dimensional (𝑛≥1) subsets, Topology 18 (1979), no. 1, 91–95. MR 528239, https://doi.org/10.1016/0040-9383(79)90017-X