Constructing a coarse space with a given Higson or binary corona

Abstract

For any compact Hausdorff space K we construct a canonical finitary coarse structure EX,K on the set X of isolated points of K. This construction has two properties:(1)If a finitary coarse space (X,E) is metrizable, then its coarse structure E coincides with the coarse structure EX,K generated by the Higson compactification K of X.(2)A compact Hausdorff space K coincides with the Higson compactification of the coarse space (X,EX,K) if the set X is dense in K and the space K is Fréchet-Urysohn. This implies that a compact Hausdorff space K is homeomorphic to the Higson corona of some finitary coarse space if one of the following conditions holds: (i) K is perfectly normal; (ii) K has weight w(K)≤ω1 and character χ(K)<p. Under CH every (zero-dimensional) compact Hausdorff space of weight ≤ω1 is homeomorphic to the Higson (resp. binary) corona of some cellular finitary coarse space.