Heine–Borel theorem

In real analysis in mathematics, the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states:

For a subset ${\displaystyle S}$ of Euclidean space ${\displaystyle \mathbb {R} ^{n}}$, the following two statements are equivalent:

• ${\displaystyle S}$ is compact, that is, every open cover of ${\displaystyle S}$ has a finite subcover
• ${\displaystyle S}$ is closed and bounded.

The theorem is sometimes also called the Borel–Lebesgue lemma.