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!

Subspaces have to be closed under more general linear combinations than just $x + y$. That is, you have to have 
$$
c_1 x + c_2 y \in S
$$
whenever $x,y \in S$ for any $c_1,c_2 \in \mathbb R$ (since the original vector space is a real-vector space, the scalars for linear combinations are, in general, real numbers)
