Heine-Borel Theorem
From AoPSWiki
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.
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