Compactness

Hein-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.
• A subcover is finite if it contains only finitely many sets.