Example of a subset of $\mathbb{R}^2$ that is closed under vector addition, but not closed under scalar multiplication? I've found several examples which are closed under scalar multiplication, but not vector addition, but I can't come up with one that is closed under vector addition, but not scalar multiplication. 

The subset $S=\{(n,0)\in\mathbb{R}^2:n\in\mathbb{Z}\}$ is closed under addition but not scalar multiplication, e.g., $\frac{1}{2}\cdot (1,0)=(\frac{1}{2},0)\notin S$.
