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Extending Maps in Hilbert Manifolds

Niemiec P.

Bulletin of the Polish Academy of Sciences. Mathematics

2012

Vol. 60, no 3

295--306

Certain results on extending maps taking values in Hilbert manifolds by maps which are close to being embeddings are presented. Sufficient conditions on a map under which it is extendable by an embedding are given. In particular, it is shown that if X is a completely metrizable space of topological weight not greater than α≥ℵ0, A is a closed set in X and f:X→M is a map into a manifold M modelled on a Hilbert space of dimension α such that f(X∖A)∩f(∂A)=∅, then for every open cover U of M there is a map g:X→M which is U-close to f (on X), coincides with f on A and is an embedding of X∖A into M. If, in addition, X∖A is a connected manifold modelled on the same Hilbert space as M, and f(∂A) is a Z-set in M, then the above map g may be chosen so that g|X∖A be an open embedding.