Difference between revisions of "Heine-Borel Theorem"

Revision as of 16:27, 15 February 2008

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

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.

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-The '''Heine-Borel theorem''' is an important theorem in elementary [[Topology]].
+The '''Heine-Borel theorem''' is an important theorem in elementary [[topology]].
 ==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}}
 [[Category:Topology]]