# Real Analysis/Compact Sets

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

## Definition

editLet (*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 . 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

editLet *A* be a compact set in with usual metric, then *A* is closed and bounded.

## Theorem (Heine-Borel)

editIf , with the usual metric, then every closed and bounded subset of *X* is compact.