Dense sigma-compact subsets of infinite-dimensional manifolds

Author:
T. A. Chapman

Journal:
Trans. Amer. Math. Soc. **154** (1971), 399-426

MSC:
Primary 57.55

DOI:
https://doi.org/10.1090/S0002-9947-1971-0283828-7

MathSciNet review:
0283828

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper four classes of separable metric infinite-dimensional manifolds are studied; those which are locally the countable infinite product of lines, those which are locally open subsets of the Hubert cube, and those which are locally one of two dense sigma-compact subsets of the Hilbert cube. A number of homeomorphism, product, characterization, and embedding theorems are obtained concerning these manifolds.

**[1]**R. D. Anderson,*Hilbert space is homeomorphic to the countable infinite product of lines*, Bull. Amer. Math. Soc.**72**(1966), 515–519. MR**190888**, https://doi.org/10.1090/S0002-9904-1966-11524-0**[2]**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**205212**, https://doi.org/10.1090/S0002-9947-1967-0205212-3**[3]**R. D. Anderson,*On topological infinite deficiency*, Michigan Math. J.**14**(1967), 365–383. MR**214041****[4]**R. D. Anderson,*Strongly negligible sets in Fréchet manifolds*, Bull. Amer. Math. Soc.**75**(1969), 64–67. MR**238358**, https://doi.org/10.1090/S0002-9904-1969-12146-4**[5]**-,*A characterization of apparent boundaries of the Hilbert cube*, Notices Amer. Math. Soc.**16**(1969), 429. Abstract #697-G17.**[6]**-,*On sigma-compact subsets of infinite-dimensional spaces*, Trans. Amer. Math. Soc. (submitted).**[7]**R. D. Anderson and R. H. Bing,*A complete elementary proof that Hilbert space is homeomorphic to the countable infinite product of lines*, Bull. Amer. Math. Soc.**74**(1968), 771–792. MR**230284**, https://doi.org/10.1090/S0002-9904-1968-12044-0**[8]**R. D. Anderson and John D. McCharen,*On extending homeomorphisms to Fréchet manifolds*, Proc. Amer. Math. Soc.**25**(1970), 283–289. MR**258064**, https://doi.org/10.1090/S0002-9939-1970-0258064-5**[9]**R. D. Anderson and R. Schori,*Factors of infinite-dimensional manifolds*, Trans. Amer. Math. Soc.**142**(1969), 315–330. MR**246327**, https://doi.org/10.1090/S0002-9947-1969-0246327-5**[10]**R. D. Anderson, David W. Henderson, and James E. West,*Negligible subsets of infinite-dimensional manifolds*, Compositio Math.**21**(1969), 143–150. MR**246326****[11]**William Barit,*Small extensions of small homeomorphisms*, Notices Amer. Math. Soc.**16**(1969), 295. Abstract #663-715.**[12]**C. Bessaga and A. Pełczyński,*Estimated extension theorem, homogeneous collections and skeletons, and their applications to topological classifications of linear metric spaces and convex sets*, Fund. Math. (submitted).**[13]**T. A. Chapman,*Infinite deficiency in Fréchet manifolds*, Trans. Amer. Math. Soc.**148**(1970), 137–146. MR**256418**, https://doi.org/10.1090/S0002-9947-1970-0256418-9**[14]**David W. Henderson,*Infinite-dimensional manifolds are open subsets of Hilbert space*, Bull. Amer. Math. Soc.**75**(1969), 759–762. MR**247634**, https://doi.org/10.1090/S0002-9904-1969-12276-7**[15]**Witold Hurewicz and Henry Wallman,*Dimension Theory*, Princeton Mathematical Series, v. 4, Princeton University Press, Princeton, N. J., 1941. MR**0006493****[16]**K. Kuratowski,*Topologie*. Vol. 1:*Espaces métrisables, espaces complets*, 2nd ed., Monografie Mat., Tom 20, PWN, Warsaw, 1948. MR**10**, 389.**[17]**H. Toruńczyk,*Skeletonized sets in complete metric spaces and homeomorphisms of the Hilbert cube*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.**18**(1970), 119–126 (English, with Russian summary). MR**264602****[18]**James E. West,*Infinite products which are Hilbert cubes*, Trans. Amer. Math. Soc.**150**(1970), 1–25. MR**266147**, https://doi.org/10.1090/S0002-9947-1970-0266147-3

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1971-0283828-7

Keywords:
Fréchet manifold,
Hubert cube manifold,
compact absorption property,
Property Z,
infinite deficiency,
finite-dimensional compact absorption property

Article copyright:
© Copyright 1971
American Mathematical Society