Higson compactifications of Wallman type (Q2631606)

This paper considers the question: is every Higson compactification of Wallman type? \par A compactification is of Wallman type if it is equivalent to some Wallman compactification, and the Higson compactification of a proper metric space $X$ is defined by embedding the space $X$ in a Tychonoff product with respect to the set $C_h(X)$ of all slowly oscillating functions. \par The main result of the paper is to show that for every proper metric space $X$, if $X$ has property HW (see below), then the Higson compactification of $X$ is of Wallman type. Here, for each $R>0$, a metric space $(X,d)$ is said to have property HW($R$) if the following condition holds: $\forall \mathcal{F}^\prime\subset\mathcal{F} \, (\bigcap\mathcal{F}^\prime = \emptyset \Rightarrow \exists F, F^\prime\in \mathcal{F}^\prime \, (d(F, F^\prime)\geq R))$. A metric space $X$ is said to have property HW if for each $R>0$, there exists a boundedly finite uniformly bounded cover $\mathcal{F}$ of $X$ with property HW($R$). The authors give some examples of metric spaces having property HW. For example, every proper metric space with asymptotic dimension at most 1 has property HW. They also show that property HW is a coarse invariant.