 Note that compactness depends only on the topology, while boundedness depends on the metric.Lebesgue's number lemma states that for every open cover of a compact metric space "M", there exists a "Lebesgue number" δ such that every subset of "M" of diameter < δ is contained in some member of the cover.Every compact metric space is second countable, and is a continuous image of the Cantor set.