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Determine whether or not the distance between nonempty $A,B\subset X$ for metric space $X$ is assumed if A and B are closed.

The definition of distance between sets A and B is $d(A,B)=\inf\{d(a,b)|a\in A, b\in B\}$ and the distance is "assumed" if $\exists a_o\in A, b_o\in B$ : $d(A,B)=d(a_o,b_o)$.

I am trying to see how it could be the case that $A,B$ closed $\implies$ $d(A,B)$ assumed. Is it because closed sets contain all their accumulation points?

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It is not the case. Consider the two curves in $\mathbf{R}^2$, $A=\{(x,-1/x)\mid x<0\}$ and $B=\{(x,1/x)\mid x>0\}$. $d(A,B)=0$ but $d(u,v)>0$ for all $u\in A$ and $v \in B$.

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    @CarlMummert : does it work if $A$ is compact, $B$ is closed, and $X$ is complete?2013-10-01