
Open  Close Function.
Hello, Im trying to prove the following function is open but not close. I really have no idea how to do it. I just know the concept but nothing else. Please help.
$\displaystyle $\pi _1 :\mathbb{R}^2 \to \mathbb{R} \hfill \\$$
$\displaystyle $\pi _1 \left( {x_1 ,x_2 } \right) = x_1 \hfill \\ $$

Take $\displaystyle E:=\left\{\left(x,\frac 1x\right),x>0\right\}$. This is a closed subset of $\displaystyle \mathbb{R}^2$. What is $\displaystyle \pi_1(E)$ ?
Take $\displaystyle U$ an open subset of $\displaystyle \mathbb R^2$ and $\displaystyle x\in \pi(U)$. Let$\displaystyle (x,y')$ such that $\displaystyle \pi (x,y)\in U$. We can find $\displaystyle I_1$ and $\displaystyle I_2$ two open intervals such that $\displaystyle (x,y)\in I_1\times I_2 \subset U$.