Deficiency in $F$-manifolds

Author:
William H. Cutler

Journal:
Proc. Amer. Math. Soc. **34** (1972), 260-266

MSC:
Primary 58B05

DOI:
https://doi.org/10.1090/S0002-9939-1972-0298710-5

MathSciNet review:
0298710

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Abstract | References | Similar Articles | Additional Information

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.

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Keywords:
Infinite-dimensional manifold,
deficiency,
negligibility,
variable product

Article copyright:
© Copyright 1972
American Mathematical Society