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Let $(X,d)$ be a metric space, $F$ included in $X$ and closed, p a point in $X-F$.

Prove that if any closed ball in X is compact then there exists y in F such that

$d(p,y)=d(p,F)$.

I have been trying to do it by sequences, but i dont know how to use the hypothesis.

JohnD
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Raiden
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  • Isn't $d(p,X)$ going to be zero? Do you mean $d(p,F)$? – Dom Dec 02 '14 at 04:24
  • Get us started by showing us your thoughts and what you have tried. Then we will likely chime in to help. – JohnD Dec 02 '14 at 04:27
  • Your idea with sequences is on the right path. Basically, the distance $d(p,F)$ is the infimum over $x\in F$ of $d(p,x)$. If you know (or can see that $d(p,\cdot)$ is a continuous function, then what can you say about the infimum over a compact set (can you see that $p$ must be in some compact set?)? – Dom Dec 02 '14 at 04:36

2 Answers2

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Pick some $x\in F$ and let $\alpha=d(p,x)$, then we can restrict our search to $\overline{B_\alpha(p)}\cap F$ (where $B_r(a)$ is the ball of radius $r$ around $a$). This is a compact set as it is a closed subset of a compact set. Now $d(p,\cdot)$ is continuous and hence its infimum must be attained over $\overline{B_\alpha(p)}\cap F$. So there is some $y\in F$ that get us the result.

Dom
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You can see here

A metric space such that all closed balls are compact is complete.

that every metric space in which the balls are compact is complete.

Then construct a sequence $x_n$ in $F$ such that $d(p,x_n)-d(p,X)<\displaystyle\frac{1}{2n}$. Prove that this sequence is Cauchy.

Sorry by the mistake before!

Community
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EQJ
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