V.I. Arnold's conjecture on avoidance of intersection, Hausdorff codimension, and a structure theorem for finite index maps

The present work solves V.I. Arnold's conjecture on avoidance of intersection. The solution depends on determining the size of the image of a finite index map between infinite dimensional separable Hilbert spaces. We define the Hausdorff codimension of subsets of infinite dimensional linear space. Hausdorff codimension measures the metric size of subsets of an infinite dimensional linear space, and includes previous descriptions by B. White and others on the size of sets. We apply Hausdorff codimension to finite index maps and establish a structure theorem for these maps that includes Smale's extension of Sard's Theorem. An application of the structure theorem to the avoidance of intersection problem determines the metric size of a set of functions whose images intersect a given countable union of smooth submanifolds each with codimension at least two.