question about topology in R^2

**bha** · March 27th 2013, 05:47 PM

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

**hollywood** · March 27th 2013, 11:33 PM

How about the graph G of the equation $y=\ln{x}$ . It's complement $G^C$ is an open set in $\mathbb{R}^2$ : if you choose a point P not on the curve, it is a distance $\epsilon$ from the curve, so the ball of radius $\frac{\epsilon}{2}$ centered at P is contained in $G^C$ . So G is closed.

But its projection onto the x-axis is the interval $(0,\infty)$ , which is an open interval.

Is that what you were looking for?

- Hollywood

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