Borel-Lebesgue covering theorem

Let $A$ be a bounded closed set in $\mathbf R^n$ and let $G$ be an open covering of it, i.e. a system of open sets the union of which contains $A$; then there exists a finite subsystem of sets $\{G_i\}$, $i=1,\ldots,N$, in $G$ (a subcovering) which is also a covering of $A$, i.e.

$$A\subset\bigcup_{i=1}^NG_i.$$

The Borel–Lebesgue theorem has a converse: If $A\subset\mathbf R^n$ and if a finite subcovering may be extracted from any open covering of $A$, then $A$ is closed and bounded. The possibility of extracting a finite subcovering out of any open covering of a set $A$ is often taken to be the definition of the set $A$ to be compact. According to such a terminology, the Borel–Lebesgue theorem and the converse theorem assume the following form: For a set $A\subset\mathbf R^n$ to be compact it is necessary and sufficient for $A$ to be bounded and closed. The theorem was proved in 1898 by E. Borel [1] for the case when $A$ is a segment $[a,b]\subset\mathbf R^1$ and $G$ is a system of intervals; the theorem was given its ultimate form by H. Lebesgue [2] in 1900–1910. Alternative names for the theorem are Borel lemma, Heine–Borel lemma, Heine–Borel theorem.

References

[1]	E. Borel, "Leçons sur la théorie des fonctions" , Gauthier-Villars (1928)
[2]	W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953)