If S+K is compact in the complex plane , are K and S compact? – math.stackexchange.com

If S+K is compact in the complex plane , are K and S compact? – math.stackexchange.com

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Let $K$ and $S$ two subsest of the complex plane such that $K+S$ is compact. Are $K$ and $S$ compact? Where $$ K+S=\{k+s: k\in K,~s\in S\}$$

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