View source for Borel-Lebesgue covering theorem

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017100/b0171001.png" /> be a bounded closed set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017100/b0171002.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017100/b0171003.png" /> be an open covering of it, i.e. a system of open sets the union of which contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017100/b0171004.png" />; then there exists a finite subsystem of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017100/b0171005.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017100/b0171006.png" />, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017100/b0171007.png" /> (a subcovering) which is also a covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017100/b0171008.png" />, i.e.

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017100/b0171009.png" /></td> </tr></table>

The Borel–Lebesgue theorem has a converse: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017100/b01710010.png" /> and if a finite subcovering may be extracted from any open covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017100/b01710011.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017100/b01710012.png" /> is closed and bounded. The possibility of extracting a finite subcovering out of any open covering of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017100/b01710013.png" /> is often taken to be the definition of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017100/b01710014.png" /> to be compact. According to such a terminology, the Borel–Lebesgue theorem and the converse theorem assume the following form: For a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017100/b01710015.png" /> to be compact it is necessary and sufficient for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017100/b01710016.png" /> to be bounded and closed. The theorem was proved in 1898 by E. Borel [[#References|[1]]] for the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017100/b01710017.png" /> is a segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017100/b01710018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017100/b01710019.png" /> is a system of intervals; the theorem was given its ultimate form by H. Lebesgue [[#References|[2]]] in 1900–1910. Alternative names for the theorem are Borel lemma, Heine–Borel lemma, Heine–Borel theorem.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Borel,   "Leçons sur la théorie des fonctions" , Gauthier-Villars  (1928)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Rudin,   "Principles of mathematical analysis" , McGraw-Hill  (1953)</TD></TR></table>