How many miles to $βω$? -- Approximating $βω$ by metric-dependent compactifications

Abstract

It is known that the Stone-Čech compactification of a non-compact metrizable space $X$ is approximated by the collection of Smirnov compactifications of $X$ for all compatible metrics on $X$. We investigate the smallest cardinality of a set $D$ of compatible metrics on the countable discrete space $ω$ such that, the Stone-Čech compactification of $ω$ is approximated by Smirnov compactifications for all metrics in $D$, but any finite subset of $D$ does not suffice. We also study the corresponding cardinality for Higson compactifications.