# Real Analysis/Compact Sets

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## Definition[edit]

Let (*X*, *d*) be a metric space and let *A* ⊆ *X*. A collection of open sets {*U*_{λ}}_{λ∈Λ} is called an open cover of if A ⊆ ∪_{λ∈Λ} *U*_{λ}.

## Definition[edit]

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 . 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[edit]

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

## Theorem (Heine-Borel)[edit]

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