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?
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Are you talking about two classes: the one determined by $1$ and the other by $-1$? So, if you identify $x\sim \lambda x$, $\lambda >0$ you will obtain the sphere $S^0$. – Sigur Jul 02 '13 at 16:30
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Sorry about the terminology, I edited the question to be clear on what I am trying to say: if we start from an affine space of dimension $n+1$, and then quotient it out by action of positive real numbers only, is it going to work? – user22835 Jul 02 '13 at 16:53
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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...
Emanuele Paolini
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