Real Analysis/Compact Sets

Compact sets are sets for which every open cover, or sequence of open sets that contain , contains a finite cover of . Definition Let (X, d) be a metric space and let A ⊆  X. We say that A is compact if for every open cover {Uλ}λ∈Λ there is a finite collection Uλ1, …,Uλk so that . In other words a set is compact if and only if every open cover has a finite subcover. There is also a sequential definition of compact set. A set A in the metric space X is called compact if every sequence in that set has a convergent subsequence. Theorems about compact sets Theorem Every infinite subset K of a compact set V has a limit point in V, as otherwise there would exist an infinite sequence of neighborhoods around points of K that does not contain a finite cover of V. Theorem 2 Every closed subset of a compact set is compact. Proof: If is closed, and , then is open, so the union of any open cover of and is an open cover of . Therefore, must contain a finite subcover of , and contains a finite subcover of . Theorem (Heine-Borel) If , with the usual metric, then every closed and bounded subset of X is compact.