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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Infinite deficiency in Fréchet manifolds


Author: T. A. Chapman
Journal: Trans. Amer. Math. Soc. 148 (1970), 137-146
MSC: Primary 57.55; Secondary 54.00
DOI: https://doi.org/10.1090/S0002-9947-1970-0256418-9
MathSciNet review: 0256418
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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.


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Keywords: The Hilbert cube, Fr&#233;chet manifolds, Property <I>Z</I>, infinite deficiency
Article copyright: © Copyright 1970 American Mathematical Society