Let $(X,d)$ be a metric space and for $S \subseteq X$, $\Gamma \ge 0$, define the $\Gamma$-dilation of $S$ to be
$$S + \Gamma := \{x \in X : \exists s \in S \text{ such that } d(x,s) \le \Gamma\}.$$
If $S$ is compact then it's easy to see that $S + \Gamma$ is closed because, by the extreme value theorem, for any $x \notin S+\Gamma$ the function $S \ni s \mapsto d(x,s)$ has an absolute minimum $\Gamma_0 > \Gamma$.
My question is, is $S + \Gamma$ still closed if $S$ is only known to be closed, or is there a counterexample?
Any hints or solutions are greatly appreciated!