open subset of manifold

Question

I saw a conclusion: If $U$ is an open subset of $\mathbb{R}^{n^2}$, then $U$ has dimension $n^2$.

My quesition : if $n=2, U=\{(x,y,0,0):x,y \in \Bbb R\}$, $U$ has a basis $(1,0,0,0),(0,1,0,0)$, so its dimension is 2.

Where is wrong?