Namespaces
Variants
Actions

Difference between revisions of "Borel-Lebesgue covering theorem"

From Encyclopedia of Mathematics
Jump to: navigation, search
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
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.
+
{{TEX|done}}
 +
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.
  
<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>
+
$$A\subset\bigcup_{i=1}^NG_i.$$
  
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.
+
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 [[#References|[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 [[#References|[2]]] in 1900–1910. Alternative names for the theorem are Borel lemma, Heine–Borel lemma, Heine–Borel theorem.
  
 
====References====
 
====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>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  E. Borel,  "Leçons sur la théorie des fonctions" , Gauthier-Villars  (1928) {{ZBL|54.0327.02}}</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  W. Rudin,  "Principles of mathematical analysis" , McGraw-Hill  (1953)</TD></TR>
 +
</table>

Latest revision as of 14:28, 19 March 2023

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) Zbl 54.0327.02
[2] W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953)
How to Cite This Entry:
Borel-Lebesgue covering theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Borel-Lebesgue_covering_theorem&oldid=22169
This article was adapted from an original article by I.A. Vinogradova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article