# Open - Close Function.

• June 9th 2011, 10:55 AM
liquidpaper
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.

$\pi _1 :\mathbb{R}^2 \to \mathbb{R} \hfill \\$
$\pi _1 \left( {x_1 ,x_2 } \right) = x_1 \hfill \\$
• June 9th 2011, 11:37 AM
girdav
Take $E:=\left\{\left(x,\frac 1x\right),x>0\right\}$. This is a closed subset of $\mathbb{R}^2$. What is $\pi_1(E)$ ?
Take $U$ an open subset of $\mathbb R^2$ and $x\in \pi(U)$. Let $(x,y')$ such that $\pi (x,y)\in U$. We can find $I_1$ and $I_2$ two open intervals such that $(x,y)\in I_1\times I_2 \subset U$.