# Difference between revisions of "Heine-Borel Theorem"

(New page: The '''Heine-Borel theorem''' is an important theorem in elementary Topology. ==Statement== Let <math>X</math> be a metric space Let <math>E\subset X</math> Then (1) <m...) |
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− | The '''Heine-Borel theorem''' is an important theorem in elementary [[ | + | The '''Heine-Borel theorem''' is an important theorem in elementary [[topology]]. |

==Statement== | ==Statement== | ||

− | Let <math> | + | 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]]. |

− | + | 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: | |

− | Then | + | 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]]. |

− | + | 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. | |

− | |||

+ | ==See Also== | ||

{{stub}} | {{stub}} | ||

[[Category:Topology]] | [[Category:Topology]] | ||

+ | [[Category: Theorems]] |

## Latest revision as of 12:30, 9 April 2019

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

## Statement

Let be any subset of . Then is compact if and only if is closed and bounded.

This statement does *not* hold if is replaced by an arbitrary metric space . However, a modified version of the theorem does hold:

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

In the totally bounded sets are precisely the bounded sets, so this new formulation does indeed imply the original theorem.

## See Also

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