The problem is: For compact metric space $(X,d)$ prove that for every $r>0$ there exists a subset $S$ of $X$ such that $\{\mbox{Open balls of radius }r\mbox{ centered at }p \mid\mbox{ for all }p \in S\}$ forms a cover for $X$ and for every $p,q \in S$, $d(p,q) > r/2$.
I have an algorithm that would go like this: Form an open cover of $X$ by the set of open balls of radius $r/2$ around all points in $X$. By compactness there exists a finite number of these balls which cover $X$. Then for each point with a ball around it, "delete" the points which are inside of the ball and not the center of the ball. Then you will have a collection of points that are at least $r/2$ distance apart and the set of balls of radius $r$ around these points will cover $X$.
Does this "algorithm" work, and if so how do you denote such a set? I'm having problems figuring out exactly how to "delete" as I've used the word above.
Thanks!