Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T07:43:22.696Z Has data issue: false hasContentIssue false
  • English
  • Français

Unions of rectifiable curves in Euclidean space and the covering number of the meagre ideal

Published online by Cambridge University Press:  12 March 2014

Juris Steprāns
Juris Steprāns*
Affiliation:
Department of Mathematics, York University, 4700 Keele Street, North York, Ontario, CanadaM3J 1P3, E-mail: steprans@mathstat.yorku.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

To any metric space it is possible to associate the cardinal invariant corresponding to the least number of rectifiable curves in the space whose union is not meagre. It is shown that this invariant can vary with the metric space considered, even when restricted to the class of convex subspaces of separable Banach spaces. As a corollary it is obtained that it is consistent with set theory that any set of reals of size ℵ1 is meagre yet there are ℵ1 rectifiable curves in ℝ3 whose union is not meagre. The consistency of this statement when the phrase “rectifiable curves” is replaced by “straight lines” remains open.

Keywords

meagre setrectifiable curveproper forcing
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 2 , June 1999 , pp. 701 - 726
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] Bartoszyński, Tomek and Judah, Haim, Set theory, A. K. Peters, Ltd., Wellesley, Massachusetts, 1995, On the structure of the real line.CrossRefGoogle Scholar
[2] Cichon, J., RosŁonowski, A., Steprāns, J., and Wȩglorz, B., Combinatorial properties of the ideal , this Journal, vol. 58 (1993), no. 1, pp. 42–54.Google Scholar
[3] Cichoń, Jacek and Morayne, Michał, On differentiability of Peano type functions, III, Proceedings of the American Mathematical Society, vol. 92 (1984), no. 3, pp. 432–438.CrossRefGoogle Scholar
[4] Miller, A., Questions 1996, see http://www.math.ufl.edu/˜logic/.Google Scholar
[5] Morayne, M., On differentiability of Peano type functions, Colloquium Mathematicum, vol. 48 (1984), no. 2, pp. 261–264.Google Scholar
[6] Morayne, Michał, On differentiability of Peano type functions I, II, Colloquium Mathematicum, vol. 53 (1987), no. 1, pp. 129–132, 133–135.Google Scholar
[7] Shelah, Saharon, On cardinal invariants of the continuum, Proceedings of the AMS-IMS-S1AM joint summer research conference held in Boulder, Colorado, June 19–25, 1983, Contemporary Mathematics, vol. 31, American Mathematical Society, Providence, Rhode Island, 1984, pp. 183–207.Google Scholar
[8] Shelah, Saharon, Vive la différence, I: Nonisomorphism of ultrapowers of countable models, Set theory of the continuum (Berkeley, California, 1989), Mathematical Sciences Research Institute Publications, vol. 26, Springer-Verlag, New York, 1992, pp. 357–405.Google Scholar
[9] Steprāns, J., Decomposing with smooth sets, Transactions of the American Mathematical Society.Google Scholar
[10] Steprāns, J., Almost disjoint families of lattice paths, Topology Proceedings, vol. 16 (1991), pp. 185–200.Google Scholar
[11] Steprāns, J., Cardinal invariants associated with Hausdorff capacities, Proceeding of the BEST conferences 1–3, Contemporary Mathematics, vol. 192, American Mathematical Society, 1995, pp. 157–184.Google Scholar
[12] Steprāns, J. and Watson, W. S., Homeomorphisms of manifolds with prescribed behaviour on large dense sets, Bulletin of the London Mathematical Society, vol. 19 (1987), pp. 305–310.Google Scholar