Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-25T19:06:29.046Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite powers of strong measure zero sets

Published online by Cambridge University Press:  12 March 2014

Marion Scheepers
Marion Scheepers*
Affiliation:
Department of Mathematics, Boise State University, 1910 University Drive Boise, Idaho 8372 5, USA E-mail: mscheep@micron.net
Get access Rights & Permissions [Opens in a new window]

Abstract

In a previous paper—[17]—we characterized strong measure zero sets of reals in terms of a Ramseyan partition relation on certain subspaces of the Alexandroff duplicate of the unit interval. This framework gave only indirect access to the relevant sets of real numbers. We now work more directly with the sets in question, and since it costs little in additional technicalities, we consider the more general context of metric spaces and prove:

1. If a metric space has a covering property of Hurewicz and has strong measure zero, then its product with any strong measure zero metric space is a strong measure zero metric space (Theorem 1 and Lemma 3).

2. A subspace X of a σ-compact metric space Y has strong measure zero if, and only if, a certain Ramseyan partition relation holds for Y (Theorem 9).

3. A subspace X of a σ-compact metric space Y has strong measure zero in all finite powers if, and only if, a certain Ramseyan partition relation holds for Y (Theorem 12).

Then 2 and 3 yield characterizations of strong measure zeroness for σ-totally bounded metric spaces in terms of Ramseyan theorems.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 3 , September 1999 , pp. 1295 - 1306
Copyright
Copyright © Association for Symbolic Logic 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Bokel, E., Sur la classification des ensembles de mesure nulle, Bulletin de la Societe Mathematique de France, vol. 47 (1919), pp. 97–125.Google Scholar
[2] Fremlin, D. H. and Miller, A. W., On some properties of Hurewicz, Menger, and Rothberger, Fundamenta Mathematicae, vol. 129 (1988), pp. 17–33.Google Scholar
[3] Galvin, F. and Miller, A. W., γ-sets and other singular sets of real numbers, Topology and its Applications, vol. 17 (1984), pp. 145–155.Google Scholar
[4] Galvin, F., Mycielski, J., and Solovay, R. M., Abstract A-280, Notices of the American Mathematical Society, vol. 26 (1979).Google Scholar
[5] Gerlits, J. and Nagy, Zs., Some properties of C(X), I, Topology and its Applications, vol. 14 (1982), pp. 151–161.Google Scholar
[6] Goldstern, M., Judah, H., and Shelah, S., Strong measure zero sets without Cohen reals, this Journal, vol. 58 (1993), pp. 1323–1341.Google Scholar
[7] Hurewicz, W., Über eine Verallgemeinerung des Boreischen Theorems, Mathematische Zeitschrift, vol. 24 (1925), pp. 401–421.Google Scholar
[8] Just, W., Miller, A. W., Scheepers, M., and Szeptycki, P. J., The combinatorics of open covers II, Topology and its Applications, vol. 73 (1996), pp. 241–266.Google Scholar
[9] Laver, R., On the consistency of the Borel conjecture, Acta Mathematicae, vol. 137 (1976), pp. 151–169.CrossRefGoogle Scholar
[10] Nowik, A., Scheepers, M., and Weiss, T., The algebraic sum of sets of real numbers with strong measure zero sets, this Journal, vol. 63 (1998), pp. 301–324.Google Scholar
[11] Pawlikowski, J., Undetermined sets of point-open game, Fundamenta Mathematicae, vol. 144 (1994), pp. 279–285.Google Scholar
[12] Rothberger, F., Eine Verschärfung der Eigenschaft C, Fundamenta Mathematicae, vol. 30 (1938), pp. 50–55.CrossRefGoogle Scholar
[13] Rothberger, F., Sur les familles indénombrables de suites de nombres naturels at les problèmes concernant la propriété C, Proceedings of the Cambridge Philosophical Society, vol. 37 (1941), pp. 109–126.Google Scholar
[14] Sakai, M., Property C″ and function spaces, Proceedings of the American Mathematical Society, vol. 104 (1988), pp. 917–919.Google Scholar
[15] Scheepers, M., Combinatorics of open covers I: Ramsey theory, Topology and its Applications, vol. 69 (1996), pp. 31–62.Google Scholar
[16] Scheepers, M., Combinatorics of open covers III: games, CP (X), Fundamenta Mathematicae, vol. 152 (1997), pp. 231–254.Google Scholar
[17] Scheepers, M., Combinatorics of open covers (iv): subspaces of the Alexandroff double of the unit interval, Topology and its Applications, vol. 83 (1998), pp. 63–75.Google Scholar
[18] Scheepers, M., Open covers and partition relations, Proceedings of the American Mathematical Society, vol. 127 (1999), pp. 577–581.Google Scholar
[19] Sierpiński, W., Sur un ensemble non dénombrable, donc toute image continue est de mesure nulle, Fundamenta Mathematicae, vol. 11 (1928), pp. 301–304.CrossRefGoogle Scholar
[20] Sierpiński, W., Sur le produit combinatoire de deux ensembles jouissant de la propriété C, Fundamenta Mathematicae, vol. 24 (1935), pp. 48–50.Google Scholar
[21] Willard, S., General topology, Addison-Wesley Publishing Company, 1970.Google Scholar