Quotient by the action of positive reals

Question

Today in my complex analysis class Riemann sphere was defined, and of course the construction caused questions, such as "why don't we distinguish between all the various infinities?" and "Would it work for reals, so as to distinguish between $-\infty$ and $\infty$?". So, this is the motivation for my question: can we construct "projective space" from an affine space quotiented out only by dilation, and not reflection or turning?

Answer 1

You can take the affine plane and add a point at infinity in every direction. What you get is a topological disk.

In general you can consider the map $\phi\colon \mathbb R^n \to B^n$ defined by: $$ \phi(x) = \frac{x}{1+|x|}. $$ This homeomorphism induces a metric on $\mathbb R^n$, the completion of this metric is the space you are thinking about...