Fake boundary sets in the Hilbert cube

Philip L. Bowers
1985 Proceedings of the American Mathematical Society  
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
more » ... 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).
doi:10.1090/s0002-9939-1985-0766541-4 fatcat:wgoq4le33rayrmngwz2brcvyl4