Fake Boundary Sets in the Hilbert Cube
Philip L. Bowers · Proceedings of the American Mathematical Society · 1985
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).