-sets in ANR's

Author:
David W. Henderson

Journal:
Trans. Amer. Math. Soc. **213** (1975), 205-216

MSC:
Primary 54C55; Secondary 57A20

DOI:
https://doi.org/10.1090/S0002-9947-1975-0391008-3

MathSciNet review:
0391008

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: (1) *Let A be a closed Z-set in an ANR X. Let* *be an open cover of X. Then there is a homotopy inverse* *to the inclusion* *such that f and both homotopies are limited by* .

(2) *If, in addition, X is a manifold modeled on a metrizable locally convex TVS, F, such that F is homeomorphic to* , *then there is a homotopy* *limited by* *such that the closure* (*in X*) *of* *is contained in* .

**[1]**R. D. Anderson,*On topological infinite deficiency*, Michigan Math. J.**14**(1967), 365–383. MR**0214041****[2]**R. D. Anderson, David W. Henderson, and James E. West,*Negligible subsets of infinite-dimensional manifolds*, Compositio Math.**21**(1969), 143–150. MR**0246326****[3]**Karol Borsuk,*Theory of retracts*, Monografie Matematyczne, Tom 44, Państwowe Wydawnictwo Naukowe, Warsaw, 1967. MR**0216473****[4]**T. A. Chapman,*Deficiency in infinite-dimensional manifolds*, General Topology and Appl.**1**(1971), 263–272. MR**0322898****[5]**C. H. Dowker,*Mapping theorems for non-compact spaces*, Amer. J. Math.**69**(1947), 200–242. MR**0020771**, https://doi.org/10.2307/2371848**[6]**K. H. Dauker,*Affine and euclidean complexes*, Dokl. Akad. Nauk SSSR**128**(1959), 655–656 (Russian). MR**0117708****[7]**James Dugundji,*Topology*, Allyn and Bacon, Inc., Boston, Mass., 1966. MR**0193606****[8]**James Eells Jr. and Nicolaas H. Kuiper,*Homotopy negligible subsets*, Compositio Math.**21**(1969), 155–161. MR**0253331****[9]**R. Engelking,*Outline of general topology*, Translated from the Polish by K. Sieklucki, North-Holland Publishing Co., Amsterdam; PWN-Polish Scientific Publishers, Warsaw; Interscience Publishers Division John Wiley & Sons, Inc., New York, 1968. MR**0230273****[10]**A. M. Gleason,*Spaces with a compact Lie group of transformations*, Proc. Amer. Math. Soc.**1**(1950), 35–43. MR**0033830**, https://doi.org/10.1090/S0002-9939-1950-0033830-7**[11]**Richard E. Heisey,*Manifolds modelled on 𝑅^{∞} or bounded weak-* topologies*, Trans. Amer. Math. Soc.**206**(1975), 295–312. MR**0397768**, https://doi.org/10.1090/S0002-9947-1975-0397768-X**[12]**David W. Henderson,*Corrections and extensions of two papers about infinite-dimensional manifolds*, General Topology and Appl.**1**(1971), 321–327. MR**0293677****[13]**David W. Henderson,*Stable classification of infinite-dimensional manifolds by homotopy-type*, Invent. Math.**12**(1971), 48–56. MR**0290413**, https://doi.org/10.1007/BF01389826**[14]**David W. Henderson,*Micro-bundles with infinite-dimensional fibers are trivial*, Invent. Math.**11**(1970), 293–303. MR**0282380**, https://doi.org/10.1007/BF01403183**[15]**J. L. Kelley and Isaac Namioka,*Linear topological spaces*, With the collaboration of W. F. Donoghue, Jr., Kenneth R. Lucas, B. J. Pettis, Ebbe Thue Poulsen, G. Baley Price, Wendy Robertson, W. R. Scott, Kennan T. Smith. The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J., 1963. MR**0166578****[16]**W. K. Mason,*Deficiency in spaces of homeomorphisms*(to appear).**[17]**R. Schori,*Topological stability for infinite-dimensional manifolds*, Compositio Math.**23**(1971), 87–100. MR**0287586****[18]**H. Torunczyk, (*G, K*)-*skeleton and absorbing sets in complete metric spaces*, Fund. Math. (to appear).

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
54C55,
57A20

Retrieve articles in all journals with MSC: 54C55, 57A20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1975-0391008-3

Article copyright:
© Copyright 1975
American Mathematical Society