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 {U_λ}_λ∈Λ there is a finite collection U_λ1, …,U_λk 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.