Heine Borel Theorem: a set in R is compact if and only if it is closed and bounded. Different concepts of compactness can be equivalent in a metric space. One such concept uses the idea of an open cover. Let M be a metric space and A⊆M

i) A cover of A is a family V of sets such that A ⊆ U V. V is an open cover if all sets in V are open. a subcover of the cover V is a subcollection of V whose union also contains A. subcover is finite if it contains only finitely many sets.