I have a set $\displaystyle A:=\{(x,x)|x\in[0,1]\}\subset [0,1]\times [0,1]$ i.e the diagonal of the unit square. How to show that A is closed??

By definition a set is closed if it's complement is open, so I need to show that $\displaystyle A^{c}$ is an open set. What $\displaystyle A^{c}$ will be in this case?

$\displaystyle A^{c}=\{(x,x)|x\not\in[0,1]\}$?

is $\displaystyle A^{c}=\{(x,x)|0>x>1|y=x\}=\{(x,x)|x\in(-\infty,0)\cup(0,\infty)|y=x\}$?