Basically, can you map all the points on a coordinate plane to unique points on the space above the x-axis?
The way I'm imagining it is that the vertical lines infinitely close to the right and left of the y-axis would become y=abs(1/ax) where x>0 and a approaches infinity. And a followup question is would a circle placed around the origin be an infinitely wide disk that bulges at x=0 in this new space?
If this is possible, would it also be possible to map all the points from $\mathbb{R}^2$ onto a horizontal section of 2D space with a lower bound of the x-axis and an upper bound of y=1?