Deficiency in $F$-manifolds

Abstract:

Let M be a manifold modelled on a metrizable, locally-convex, topological vector space F such that $F \cong {F^\omega }$, and let K be a closed subset of M. Then the following are equivalent: (1) K is locally a subset of a collared submanifold of M, (2) each $x \in K$ has an open neighborhood U and a homeomorphism $h:U \to {l_2} \times F$ such that $h(U \cap K) \subset \{ 0\} \times F$, (3) each $x \in K$ has an open neighborhood U and a homeomorphism $h:U \to F \times F$ such that $h(U \cap K) \subset \{ 0\} \times F$, (4) there is a homeomorphism $h:M \to M \times F$ such that for $x \in K,h(x) = (x,0)$, (5) K is infinite-deficient (i.e. there is a homeomorphism $h:M \to M \times {l_2}$ such that $h(K) \subset M \times \{ 0\} )$, and (6) K is the finite union of sets each having one of the above properties.