Let $S,T$ be 2 set. Prove that there is a $x\in S$ s.t, $d(x,T)=d(S,T)$ if $S$ is a compact set. Here, $d(S,T)$ denoted the $\inf\{d(s,t):s\in S, t\in T\}$.
proving something about the infimum distance between 2 set
3
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general-topology
metric-spaces
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0Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. Also, many find the use of imperative ("Prove", "Solve", etc.) to be rude when asking for help; please consider rewriting your post. – 2012-10-15
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0A few related/similar questions: [Continuity of the metric function](http://math.stackexchange.com/questions/48850/continuity-of-the-metric-function), [If $A$ is compact and $B$ is closed, show $d(A,B)$ is achieved](http://math.stackexchange.com/questions/109167/if-a-is-compact-and-b-is-closed-show-da-b-is-achieved) and [Prove that the supremum of the distance between a point in a compact subset and any other subset does not (sic) attain the supremum.](http://math.stackexchange.com/questions/204766/prove-that-the-supremum-of-the-distance-between-a-point-in-a-compact-subset-and) – 2012-10-18
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0is there any way to attack this problem without use of functions, limits etc? Ie is there a way to prove this using concepts of open balls, closed balls, compactness? – 2016-09-22