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Given a finite set, $X$, with a metric, $d(x,y)$ defined on it, I am interested in the following subsets:

$S_k\subseteq X$ s.t. $\forall x\in X,\exists s\in S_k:d(x,s)\geq k$

Do such constructions have a particular name I should be searching for or any nice properties?

Asaf Karagila
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Stan
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  • If I am not wrong, your sets $S_k$ are not well-defined. Do you mean that for any $k$, there exists a unique set $S_k$ satisfying $\forall x\in X,\exists s\in S_k:d(x,s)\geq k$. This is clearly false. For example, if $X$ has the discrete topology and $k=1$, then every non empty subset $S$ satisfies $\forall x\in X,\exists s\in S:d(x,s)\geq 1$. – Taladris Jun 05 '14 at 12:13
  • I think the OP means, for some fixed real $\delta>0$, we have a collection of sets $S_\delta={S_\delta^i}$ such that for all $i$, $S_\delta^i\subset X$ and $S_\delta^i$ satisfies the property that every element in $X$ is at least $\delta$ distance away from some point in $S_\delta^i$. – Dan Rust Jun 05 '14 at 12:37
  • Apologies for the confusion. I agree that there could in general be more than one $S_k$ satisfying my requirement but was wondering if there's a generic term for them and in particular what properties they have e.g. how small they can be (although your discrete metric example suggests that answer is $1$ for $k=1$). – Stan Jun 05 '14 at 15:05
  • I'm sure that there is no particular well-known terminology for such sets. If I needed plain words to characterize such a set I would say it is "a set that is not covered by any open ball of radius $k$". – Lee Mosher Jun 05 '14 at 15:17

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