Unions of Rectifiable Curves and the Dimension of Banach Spaces

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 that any set of reals of size $\aleph_1$ is meagre yet therer are $\aleph_1$ rectifiable curves in $\Reals^3$ whose union is not meagre. The consistency of this statement when the phrase ``rectifiable curves'' is replaced by ``straight lines'' remains open.