Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

A fake topological Hilbert space


Authors: R. D. Anderson, D. W. Curtis and J. van Mill
Journal: Trans. Amer. Math. Soc. 272 (1982), 311-321
MSC: Primary 57N17; Secondary 46C05, 54G20, 57N20
DOI: https://doi.org/10.1090/S0002-9947-1982-0656491-8
MathSciNet review: 656491
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We give an example of a topologically complete separable metric AR space $ X$ which is not homeomorphic to the Hilbert space $ {l^2}$, but which has the following properties:

(i) $ X$ imbeds as a convex subset of $ {l^2}$

(ii) every compact subset of $ X$ is a $ Z$-set;

(iii) $ X \times X \approx {l^2};$

(iv) $ X$ is homogeneous;

(v) $ X \approx X\backslash G$ for every countable subset $ G$.


References [Enhancements On Off] (What's this?)

  • [1] R. D. Anderson, Hilbert space is homeomorphic to the countable infinite product of lines, Bull. Amer. Math. Soc. 72 (1966), 515-519. MR 0190888 (32:8298)
  • [2] -, On topological infinite deficiency, Michigan Math. J. 14 (1967), 365-383. 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. Bennett, Countable dense homogeneous spaces, Fund. Math. 74 (1972), 189-194. MR 0301711 (46:866)
  • [5] C. Bessaga and A. Pelczyński, Selected topics in infinite-dimensional topology, PWN, Warsaw, 1975.
  • [6] T. A. Chapman, Lectures on Hilbert cube manifolds, CBMS Regional Conf. Series in Math., no. 28, Amer. Math. Soc., Providence, R. I., 1976. MR 0423357 (54:11336)
  • [7] D. W. Curtis, Hyperspaces homeomorphic to Hilbert space, Proc. Amer. Math. Soc. 75 (1979), 126-130. MR 529228 (80d:54007)
  • [8] -, Boundary sets in the Hilbert cube, in preparation. $ 9$. T. Dobrowolski and H. Toruńczyk, Separable complete ANR's admitting a group structure are Hilbert manifolds, Topology Appl. 12 (1981), 229-235.
  • [10] J. Dugundji, Topology, Allyn and Bacon, Boston, Mass., 1966. MR 0193606 (33:1824)
  • [11] R. Geoghegan (editor), Open problems in infinite-dimensional topology, Topology Proc. 4 (1979), 287-338. MR 583711 (82a:57015)
  • [12] V. L. Klee, Convex bodies and periodic homeomorphisms in Hilbert space, Trans. Amer. Math. Soc. 74 (1953), 10-43. MR 0054850 (14:989d)
  • [13] N. Kroonenberg, Pseudo-interiors of hyperspaces, Compositio Math. 32 (1976), 113-131. MR 0413109 (54:1230)
  • [14] V. T. Liem, A counter-example in $ {l^2}$-manifold theory, Proc. Amer. Math. Soc. 73 (1979), 119-120.
  • [15] H. Torunczyk, Characterizing Hilbert space topology, Fund. Math. 111 (1981), 247-262. MR 611763 (82i:57016)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57N17, 46C05, 54G20, 57N20

Retrieve articles in all journals with MSC: 57N17, 46C05, 54G20, 57N20


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1982-0656491-8
Keywords: Hilbert space, absolute retract, discrete approximation property, $ Z$-set, homogeneity, negligible subset
Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society