How many miles to βX? II — Approximations to βX versus cofinal types of sets of metrics

Kada, Tomoyasu and Yoshinobu proved that the Stone–Čech compactification of a locally compact separable metrizable space is approximated by the collection of d-many Smirnov compactifications, where d is the dominating number. By refining the proof of this result, we will show that the collection of compatible metrics on a locally compact separable metrizable space has the same cofinal type, in the sense of Tukey relation, as the set of functions from ω to ω with respect to eventually dominating order.