Why is $S = \{(x,y) \in \mathbb{R}^2 \mid \text{$x$, $y$ even integers}\}$ not a subspace of $\mathbb{R}^2$? Let $S = \{(x,y) \in \mathbb{R}^2 \mid \text{$x$, $y$ even integers}\} \subset \mathbb{R}^2$.
Why is this not a subspace of $\mathbb{R}^2$? $0$ seems to be in it ($2(0) = 0$), $x+y$ seems to be in it, and $kx$ where $k$ is an integer seems to be in it as well.  
Thanks!

It is not a subspace of $\mathbb{ R}^2$ because $kx$ has to be in it for every real $k$, not just integer $k$.
