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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Fake boundary sets in the Hilbert cube
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by Philip L. Bowers PDF
Proc. Amer. Math. Soc. 93 (1985), 121-127 Request permission

Abstract:

For each positive integer $n$, a $\sigma$-$Z$-set ${B_n}$ in the Hilbert cube ${I^\infty }$ is constructed whose complement ${s_n} = {I^\infty } - {B_n}$ is not homeomorphic to the pseudointerior $s$ of the Hilbert cube though ${s_n}$ and ${B_n}$ satisfy: (i) every compact subset of ${s_n}$ is a $Z$-set in ${s_n}$; (ii) ${s_n} \times {s_n}$ is homeomorphic to $s$; (iii) ${B_n}$ admits small maps ${I^\infty } \to {B_n}$; (iv) ${s_n}$ satisfies the discrete $n$-cells property; and (v) ${B_n}$ is locally $(n - 1)$-connected in ${I^\infty }$. It is shown that ${s_n}$ does not satisfy the discrete $(n + 1)$-cells property and thus ${B_n}$ is not a boundary set, that is, ${s_n}$ is not homeomorphic to $s$. These examples build upon an example of Anderson, Curtis, and van Mill of a fake boundary set ${B_0}$ that satisfies (i)-(iv) for $n = 0$. Their example is not a boundary set since it fails to be locally continuum-connected. The examples constructed herein show that there is a hierarchy of fake boundary sets satisfying (i)-(iv) that satisfy higher and higher orders of a strong form of local connectivity (v).
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Additional Information
  • © Copyright 1985 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 93 (1985), 121-127
  • MSC: Primary 57N20; Secondary 54F35
  • DOI: https://doi.org/10.1090/S0002-9939-1985-0766541-4
  • MathSciNet review: 766541