Random question: Is there a set-theoretic description of projective space? [closed]

I met projective space via a recent class on perspective drawing, believe it or not, but I didn't know that this was the "space" we were using. I came across a more detailed description trawling the net.

In a book on point-set topology that I bought, it describes Euclidean n-space as a field made of (sorry I don't know how to write mathematical symbols yet):

[ {n-tuples of reals}, Op("+"), Op(".") ]

So what is the equivalent set-theoretic description for projective space? I haven't been able to find one anywhere. All I've found is that basically it is constructed by taking a regular plane and adding the 'horizon' line but I want to understand mathematically what it is. Wiki page is confusing as hell. :(

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closed as too localized by Angelo, Simon Thomas, Qiaochu Yuan, J.C. Ottem, Todd Trimble♦Feb 27 '11 at 16:51

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Yes, there is, but this isn't the appropriate place to ask about it; ask at math.stackexchange.com, for example. –  Qiaochu Yuan Feb 27 '11 at 16:30
Sorry - will try stackexchange.com. thanks. –  mathmoggy Feb 27 '11 at 16:35

One definition is that the $n$-dimensional (real) projective space is the space of lines through the origin in $R^{n+1}$. Technically it is a topological quotient of $R^{n+1}$ minus the origin, by the equivalence relation $x\sim \lambda x$ for all (nonzero) vectors $x$ in $R^{n+1}$ and all (nonzero) scalars $\lambda$.