Difference between revisions of "Heine-Borel Theorem"

(This only holds for R^n, not for general metric spaces)
(Statement)
 
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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.
 
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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==See Also==
 
{{stub}}
 
{{stub}}
  
 
[[Category:Topology]]
 
[[Category:Topology]]
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[[Category: Theorems]]

Latest revision as of 12:30, 9 April 2019

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.

See Also

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