hello, i'm looking for example for a closed group in R^2, but its projection on R is not closed. any ideas?

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- Mar 27th 2013, 04:47 PMbhaquestion about topology in R^2
hello, i'm looking for example for a closed group in R^2, but its projection on R is not closed. any ideas?

- Mar 27th 2013, 10:33 PMhollywoodRe: question about topology in R^2
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