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The projective space seems to be defined via the equivalence relation x~y is x=ky for k not 0 and [tex]RP^{n} = \mathbb{R}^{n+1} - \{ 0 \} / \sim[/tex].

Therefore you can rescale any point in [tex]\mathbb{R}^{n+1}[/tex] down onto the surface of a unit sphere [tex]S^{n}[/tex] and you also have the fact antipodal points are equivalent, so you end up with [tex]RP^{n}[/tex] defined as the hemispherical surface of one half of [tex]S^{n}[/tex], but with the entire 'equator' (ie boundary of the hemisphere) included.

Why is the entire equator included, when you only need half of it? Is it to make sure tne boundary is entirely closed and allows for this hemisphere to be identitied with [tex]D^{n}[/tex], the 'circular' disk?

That all seems fairly okay to me, but then when it comes to the complex version, [tex]CP^{n}[/tex], the extension doesn't seem to be "It's the same, but in complex space", a whole new description seems to be done (which varies from place to place I've read) and even my supervisor wasn't sure about it (not that this stuff is her thing though). Is [tex]CP^{n}[/tex] just the collection of lines through the origin in [tex]\mathbb{C}^{n+1}[/tex] or is it more subtle than that now you're in complex space?

Any help would be appreciated. It's not something I desperately need to know, but it's mentioned enough times in various books I'd like to get my head around it.