Infinite deficiency in Fréchet manifolds

Abstract:

Denote the countable infinite product of lines by s, let X be a separable metric manifold modeled on s, and let K be a closed subset of X having Property Z in X, i.e. for each nonnull, homotopically trivial, open subset U of X, it is true that $U\backslash K$ is nonnull and homotopically trivial. We prove that there is a homeomorphism h of X onto $X \times s$ such that $h(K)$ projects onto a single point in each of infinitely many different coordinate directions in s. Using this we prove that there is an embedding of X as an open subset of s such that K is carried onto a closed subset of s having Property Z in s. We also establish stronger versions of these results.