Small subsets of asymptotic resemblance spaces and their Higson corona

Abstract

We define the concept of small subsets of asymptotic resemblance spaces and obtain some of their properties. In addition, we show that if the ideal of all small subsets of a metric space $X$ coincides with the ideal of all bounded subsets of $X$, then the asymptotic dimension of $X$ is equal to zero. We also show that a subset $A$ of a proper metric space $X$ is small, if and only if, the interior of the boundary of $A$ in the Higson corona of $X$ is empty.