Real Analysis/Compact Sets

Definition of compact set If any set has a open cover and containing finite subcover than it is compact

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&isin;Λ there is a finite collection Uλ 1, …,Uλ k so that $$\textstyle A\subseteq \bigcup_{i=1}^k U_{\lambda_i}$$. 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 have a convergent subsequence.

Theorem
Let A be a compact set in $$R^n$$ with usual metric, then A is closed and bounded.

Theorem (Heine-Borel)
If $$\textstyle X = \mathbb{R}^n$$, with the usual metric, then every closed and bounded subset of X is compact.