Suppose K is a nonempty closed and bounded subset of a metric space X and x $\in$ X. Show the following hypothesis fails: There is a p $\in$ K such that, for all other q $\in$ K, d(p,x) $\leq$ d(q,x). I know this hypothesis holds if K is compact, so clearly the counterexample must not come from $\mathbb{R}$ because of the Heine-Borel theorem. Much appreciated!

  • $\begingroup$ You could try $\mathbb{R}\setminus\{0\}$. $\endgroup$ – Daniel Fischer Oct 30 '13 at 13:32
  • $\begingroup$ $\mathbb{R}$ \ {0} isn't closed though since singletons are closed, so the complement is open. :/ $\endgroup$ – Jimmy Xiao Oct 30 '13 at 17:11
  • $\begingroup$ I meant just what user103254 wrote ;) $\endgroup$ – Daniel Fischer Oct 30 '13 at 21:56

I think DF suggested $\mathbb R\setminus \{0\}$ as a candidate for $X$. It is a metric space with the metric induced from $\mathbb R$.

With this $X$, try $x=-1$ and $K=\{x\in X: 0\le x\le 1\}$ (which is a closed subset of $X$).


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