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Silly Mistake August 15, 2009

Posted by putnam120 in Math Related.
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I was on a math help forum and was trying to help someone with the following problem.

Given two disjoint closed sets A and B in a metric space S, prove that there exists disjoint open sets U and V such that A\subset{U} and B\subset{V}.

In my proof I made the following mistake: Let r=\inf d(x,y) for x\in{A}, y\in{B}. Then r>0.

I should have stopped at this time, since I remember talking about something very similar to this in my analysis class.  It turns out that to gaurantee r>0 you need that at least one of A,B to be compact.

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