For a Borel action of a locally compact group on a standard Borel space $X$, it was shown by Kechris that we can construct, for a fixed compact symmetric neighborhood of the identity $K$, a Borel set which meets every orbit of the action but is spread out in the sense that no two points in the set are within a $K$ movement of eachother. We will prove that such a set can be extended to a maximal one (meaning now that every point of $X$ must be within $K$ of the set) and discuss a couple of specific applications to the complexity of equivalence relations.