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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: [email protected]
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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

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