  It is known that the Stone-\v{C}ech 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 $\omega$ such that, the Stone-\v{C}ech compactification of
$\omega$ 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.
