Vertical order in the Hilbert cube (Q1091637)

The paper addresses the problem of detecting ''tameness'' of closed subsets of the Hilbert cube that are situated with ''small'' vertical order (with respect to some factorization \(I^{\infty}\times I)\). The main result is that a compact subset \(A\) that has infinite codimension and has countable vertical order (i.e., the cardinality of \(A\cap (\{q\}\times I)\) is countable for each \(q\in I^{\infty})\) is a \(Z\)-set. The result contrasts with the situation for Euclidean space \(E^n\), \(n\geq 4\), for there are wild subsets of each dimension \(\leq n-1\) that have finite vertical order. The result parallels the situation in \(E^3\) where 1-dimensional compact subsets with countable vertical order are tame.