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Fake boundary sets in the Hilbert cube

Author: Philip L. Bowers
Journal: Proc. Amer. Math. Soc. 93 (1985), 121-127
MSC: Primary 57N20; Secondary 54F35
MathSciNet review: 766541
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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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Keywords: Boundary set in the Hilbert cube, discrete approximation property, discrete $ n$-cells property, locally $ n$-connected in $ X$
Article copyright: © Copyright 1985 American Mathematical Society

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