Difference between revisions of "Heine-Borel Theorem"

Latest revision as of 12:30, 9 April 2019

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

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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 ==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]].
+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:
+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}}
 [[Category:Topology]]
+[[Category: Theorems]]