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. 

Let $\{v_1, v_2, \ldots\}$ be any countable subset of $\Bbb R^2$ whose elements are not all 0.  Then $$\left\{\sum_{i=1}^\infty z_iv_i \middle\lvert z_i\in \Bbb Z\right\}$$ is closed under addition. But it is a countable set, and any set (except {0}) that is closed under scalar multiplication is uncountable.
