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Open subsets of $ R\sp{\infty }$ are stable


Author: Richard E. Heisey
Journal: Proc. Amer. Math. Soc. 59 (1976), 377-380
MSC: Primary 57A20; Secondary 58B05
DOI: https://doi.org/10.1090/S0002-9939-1976-0425974-0
MathSciNet review: 0425974
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Abstract: Let $ U$ be an open subset of $ {R^\infty } = \operatorname{dir} \lim {R^n}$, where $ R$ denotes the reals. We show that $ U \times {R^\infty }$ is homeomorphic to $ U$. Combined with previous work of the author we obtain the corollary that two open subsets of $ {R^\infty }$ are homeomorphic if and only if they have the same homotopy type.


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

  • [1] R. E. Heisey, Contracting spaces of maps on the countable direct limit of a space, Trans. Amer. Math. Soc. 193 (1974), 389-412. MR 0367908 (51:4150)
  • [2] -, Manifolds modelled on $ {R^\infty }$ or bounded weak-$ ^{\ast}$ topologies, Trans. Amer. Math. Soc. 206 (1975), 295-312. MR 0397768 (53:1626)
  • [3] C P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology, Ergebnisse Math. Grenzgebiete, Band 69, Springer-Verlag, New York, 1972. MR 50 #3236. MR 0350744 (50:3236)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0425974-0
Keywords: Open subset, direct limit, stability, homotopy type
Article copyright: © Copyright 1976 American Mathematical Society

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