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Transactions of the American Mathematical Society

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

 
 

 

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

(2) If, in addition, X is a manifold modeled on a metrizable locally convex TVS, F, such that F is homeomorphic to $ {F^\omega }$, then there is a homotopy $ j:X \times I \to X$ limited by $ \mathcal{F}$ such that the closure (in X) of $ j(X \times \{ 1\} )$ is contained in $ X - A$.


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  • [1] R. D. Anderson, On topological infinite deficiency, Michigan Math. J. 14 (1967), 365-383. MR 35 #4893. MR 0214041 (35:4893)
  • [2] R. D. Anderson, D. W. Henderson and J. E. West, Negligible subsets of infinite-dimensional manifolds, Compositio Math. 21 (1969), 143-150. MR 39 #7630. MR 0246326 (39:7630)
  • [3] K. Borsuk, Theory of retracts, Monografie Mat., tom 44, PWN, Warsaw, 1967. MR 35 #7306. MR 0216473 (35:7306)
  • [4] T. A. Chapman, Deficiency in infinite-dimensional manifolds, General Topology and Appl. 1 (1971), 263-272. MR 48 #1259. MR 0322898 (48:1259)
  • [5] C. H. Dowker, Mapping theorems for non-compact spaces, Amer. J. Math. 69 (1947), 200-242. MR 8, 594. MR 0020771 (8:594g)
  • [6] -, On affine and euclidean complexes, Dokl. Akad. Nauk SSSR 128 (1959), 655-656. (Russian) MR 22 #8483. MR 0117708 (22:8483)
  • [7] J. Dugundji, Topology, Allyn and Bacon, Boston, Mass., 1966. MR 33 #1824. MR 0193606 (33:1824)
  • [8] J. Eells, Jr. and N. H. Kuiper, Homotopy negligible subsets, Compositio Math. 21 (1969), 155-161. MR 40 #6546. MR 0253331 (40:6546)
  • [9] R. Engelking, Outline of general topology, PWN, Warsaw, 1965; English transl., North-Holland, Amsterdam; Interscience, New York, 1968. MR 36 #4508; 37 #5836. MR 0230273 (37:5836)
  • [10] A. M. Gleason, Spaces with a compact Lie group of transformations, Proc. Amer. Math. Soc. 1 (1950), 35-43. MR 11, 497. MR 0033830 (11:497e)
  • [11] R. Heisey, Manifolds modelled on $ {R^\infty }$ or bounded weak-$ ^\ast$ topologies, Trans. Amer. Math. Soc. 206 (1975), 295-312. MR 0397768 (53:1626)
  • [12] D. W. Henderson, Corrections and extensions of two papers about infinite-dimensional manifolds, General Topology and Appl. 1 (1971), 321-327. MR 45 #2754. MR 0293677 (45:2754)
  • [13] -, Stable classification of infinite-dimensional manifolds by homotopy-type Invent. Math. 12 (1971), 48-56. MR 44 #7594. MR 0290413 (44:7594)
  • [14] -, Micro-bundles with infinite-dimensional fibers are trivial, Invent. Math. 11 (1970), 293-303. MR 43 #8092. MR 0282380 (43:8092)
  • [15] J. Kelley, I. Namioka et al., Linear topological spaces, University Ser. in Higher Math., Van Nostrand, Princeton, N. J., 1963. MR 29 #3851. MR 0166578 (29:3851)
  • [16] W. K. Mason, Deficiency in spaces of homeomorphisms (to appear).
  • [17] R. M. Schori, Topological stability for infinite-dimensional manifolds, Compositio Math. 23 (1971), 87-100. MR 44 #4789. MR 0287586 (44:4789)
  • [18] H. Torunczyk, (G, K)-skeleton and absorbing sets in complete metric spaces, Fund. Math. (to appear).

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DOI: https://doi.org/10.1090/S0002-9947-1975-0391008-3
Article copyright: © Copyright 1975 American Mathematical Society

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