Difference between revisions of "Heine-Borel Theorem"

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(This only holds for R^n, not for general metric spaces)
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==Statement==
 
==Statement==
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]].
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Let <math>E</math> be any [[subset]] of <math>\mathbb R^n</math>.  Then <math>E</math> is [[compact set | compact]] if and only if <math>E</math> is [[closed]] and [[bounded]].
  
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This statement does ''not'' hold if <math>\mathbb R^n</math> is replaced by an arbitrary metric space <math>X</math>. However, a modified version of the theorem does hold:
  
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Let <math>X</math> be any metric space, and let <math>E</math> be a subset of <math>X</math>. Then <math>E</math> is compact if and only if <math>E</math> is closed and [[totally bounded]].
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In <math>\mathbb R^n</math> the totally bounded sets are precisely the bounded sets, so this new formulation does indeed imply the original theorem.
  
 
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[[Category:Topology]]
 
[[Category:Topology]]

Revision as of 03:17, 5 November 2009

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

Statement

Let $E$ be any subset of $\mathbb R^n$. Then $E$ is compact if and only if $E$ is closed and bounded.

This statement does not hold if $\mathbb R^n$ is replaced by an arbitrary metric space $X$. However, a modified version of the theorem does hold:

Let $X$ be any metric space, and let $E$ be a subset of $X$. Then $E$ is compact if and only if $E$ is closed and totally bounded.

In $\mathbb R^n$ the totally bounded sets are precisely the bounded sets, so this new formulation does indeed imply the original theorem.

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