Fake Boundary Sets in the Hilbert Cube
نویسنده
چکیده
For each positive integer n, a o-Z-set B„ in the Hilbert cube 7°° is constructed whose complement s„ = Ix — B„ is not homeomorphic to the pseudointerior j of the Hilbert cube though sn and B„ satisfy: (i) every compact subset of s„ is a Z-set in s„; (ii) s„ X s„ is homeomorphic to s; (iii) Bn admits small maps 7°° -» B„; (iv) s„ satisfies the discrete «-cells property; and (v) Bn is locally (n — 1)connected in /°°. It is shown that sn does not satisfy the discrete (n + l)-cells property and thus Bn is not a boundary set, that is, s„ is not homeomorphic to s. These examples build upon an example of Anderson, Curtis, and van Mill of a fake boundary set B0 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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تاریخ انتشار 2010