# 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∈Λ there is a finite collection Uλ1, …,Uλk so that ${\displaystyle \textstyle A\subseteq \bigcup _{i=1}^{k}U_{\lambda _{i}}}$ . 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 ${\displaystyle R^{n}}$  with usual metric, then A is closed and bounded.

## Theorem (Heine-Borel)

If ${\displaystyle \textstyle X=\mathbb {R} ^{n}}$ , with the usual metric, then every closed and bounded subset of X is compact.