hello, i'm looking for example for a closed group in R^2, but its projection on R is not closed. any ideas?
How about the graph G of the equation $\displaystyle y=\ln{x}$. It's complement $\displaystyle G^C$ is an open set in $\displaystyle \mathbb{R}^2$: if you choose a point P not on the curve, it is a distance $\displaystyle \epsilon$ from the curve, so the ball of radius $\displaystyle \frac{\epsilon}{2}$ centered at P is contained in $\displaystyle G^C$. So G is closed.
But its projection onto the x-axis is the interval $\displaystyle (0,\infty)$, which is an open interval.
Is that what you were looking for?
- Hollywood