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 | References | Similar Articles | Additional Information

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 is nonnull and homotopically trivial. We prove that there is a homeomorphism *h* of *X* onto such that 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.

**[1]**R. D. Anderson,*Topological properties of the Hilbert cube and the infinite product of open intervals*, Trans. Amer. Math. Soc.**126**(1967), 200-216. MR**34**#5045. MR**0205212 (34:5045)****[2]**-,*On topological infinite deficiency*, Michigan Math. J.**14**(1967), 365-383. MR**35**#4893. MR**0214041 (35:4893)****[3]**-,*Strongly negligible sets in Fréchet manifolds*, Bull. Amer. Math. Soc.**75**(1969), 64-67. MR**0238358 (38:6634)****[4]**R. D. Anderson and R. Schori,*Factors of infinite-dimensional manifolds*, Trans. Amer. Math. Soc.**142**(1969), 315-330. MR**0246327 (39:7631)****[5]**R. D. Anderson, David W. Henderson and James E. West,*Negligible subsets of infinite-dimensional manifolds*, Compositio Math.**21**(1969), 143-150. MR**0246326 (39:7630)****[6]**William Barit, Notices Amer. Math. Soc.**16**(1969), 295, Abstract #663-715.**[7]**C. Bessaga and A. Pelczyński,*Estimated extension theorem, homogeneous collections**and skeletons, and their applications to topological classifications of linear metric spaces and convex sets*, Fund. Math. (to appear). MR**0178322 (31:2580)****[8]**David W. Henderson,*Infinite-dimensional manifolds are open subsets of Hilbert space*, Bull. Amer. Math. Soc.**75**(1969), 759-762. MR**0247634 (40:898)****[9]**H. Torunczyk,*Remarks on Anderson's paper on topological infinite deficiency*, Fund. Math. (to appear) MR**0261536 (41:6149)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1970-0256418-9

Keywords:
The Hilbert cube,
Fréchet manifolds,
Property *Z*,
infinite deficiency

Article copyright:
© Copyright 1970
American Mathematical Society