Difference between revisions of "Heine-Borel Theorem"

m
Line 1: Line 1:
The '''Heine-Borel theorem''' is an important theorem in elementary [[Topology]].
+
The '''Heine-Borel theorem''' is an important theorem in elementary [[topology]].
  
 
==Statement==
 
==Statement==
Let <math>X</math> be a [[Metric space]]
+
Let <math>X</math> be any [[metric space]] and <math>E \subseteq X</math> any [[subset]].  Then <math>E</math> is [[compact set | compact]] if and only if <math>E</math> is [[closed]] and [[bounded]].
  
Let <math>E\subset X</math>
 
  
Then
 
 
(1) <math>E</math> is closed and bounded if and only if
 
 
(2) <math>E</math> is [[Compact set|compact]]
 
  
 
{{stub}}
 
{{stub}}
  
 
[[Category:Topology]]
 
[[Category:Topology]]

Revision as of 17:27, 15 February 2008

The Heine-Borel theorem is an important theorem in elementary topology.

Statement

Let $X$ be any metric space and $E \subseteq X$ any subset. Then $E$ is compact if and only if $E$ is closed and bounded.


This article is a stub. Help us out by expanding it.

Invalid username
Login to AoPS