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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
MathSciNet review: 656491
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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$.


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DOI: http://dx.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