$C_{0}$-coarse geometry of complements of Z-sets in the Hilbert cube

Abstract

Abstract. Motivated by the Chapman Complement Theorem, we construct an isomorphism between the topological category of compact Z-sets in the Hilbert cube Q and the C 0 -coarse category of their complements. The C 0 -coarse morphisms are, in this particular case, intrinsically related to uniformly continuous proper maps. Using that fact we are able to relate in a natural way some of the topological invariants of Z-sets to the geometry of their complements.

Extension properties of Stone–Čech coronas and proper absolute extensors

“…In this situation problem (B) becomes a part of a general problem of recovering properties of X in terms of its complement I τ \ X. This leads us to considerations very similar to the study carried out in [10] for τ = ω. However, there is a major difference between the metrizable (τ = ω) and non-metrizable (τ > ω) cases.…”

Canonical covers and dimension of Z-sets in the Hilbert cube

“…In particular it was asked how the dimension can be described in this context. Later, in [4] it was proved that the topological category for compact Zsets of the Hilbert cube is isomorphic to the C 0 -coarse geometry category of their complements. The C 0 -coarse structure of a metric space was introduced by Wright [11] (see also [12]).…”

“…One of the consequences of [4] is that the complements of two compact Z-sets are C 0 -coarsely equivalent if and only if those complements are uniformly homeomorphic (with respect to the metrics induced by that fixed on the Hilbert cube) and it is equivalent to the fact that the two compact Z-sets are homeomorphic.…”

“…In view of the equivalence of categories described in [4], it is natural to consider the problem of describing topological invariants of Z-sets in terms of C 0 -coarse invariants of their complements. In this way the asymptotic dimension of such a coarse structure on the complement appears to be an appropriate tool for describing, from the outside, the topological dimension of a Z-set.…”

“…In this way the asymptotic dimension of such a coarse structure on the complement appears to be an appropriate tool for describing, from the outside, the topological dimension of a Z-set. Indeed in [4] it was proved that the dimension of a Z-set can be interpreted in terms of the asymptotic dimension of the C 0 -coarse structure on its complement. See [10] for the definition of asymptotic dimension of a coarse space.…”