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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A factorization of the direct limit of Hilbert cubes


Author: Richard E. Heisey
Journal: Proc. Amer. Math. Soc. 54 (1976), 255-260
MSC: Primary 57A20
DOI: https://doi.org/10.1090/S0002-9939-1976-0418102-9
MathSciNet review: 0418102
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Abstract: We show that the countable direct limit of Hilbert cubes $ {Q^\infty }$ is homeomorphic to the product of the Hilbert cube $ Q$ and the countable direct limit of lines $ {R^\infty }$. As a consequence, two open subsets of $ {R^\infty }$ have the same homotopy type if and only if their products with $ Q$ are homeomorphic. Combined with a theorem of D. W. Henderson our result implies that $ X \times Q \times {R^\infty } \cong Q \times {R^\infty }$, where $ X$ is any locally compact, separable AR (metric).


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DOI: https://doi.org/10.1090/S0002-9939-1976-0418102-9
Keywords: Hilbert cube, direct limit, absolute (neighborhood) retract, bounded weak-$ ^{\ast}$ topology
Article copyright: © Copyright 1976 American Mathematical Society