Open - Close Function.

**liquidpaper** · June 9th 2011, 09:55 AM

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.

$\pi _1 :\mathbb{R}^2 \to \mathbb{R} \hfill \\$
$\pi _1 \left( {x_1 ,x_2 } \right) = x_1 \hfill \\$

**girdav** · June 9th 2011, 10:37 AM

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 $(x,y)\in I_1\times I_2 \subset U$ .

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