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I've just completed exercise 2.14 in the first chapter of Hartshorne's Algebraic Geometry. This exercise asks for a proof that the image of the Segre embedding is a projective variety. The Segre embedding is the map $\mathbb{P}^r \times \mathbb{P}^s \rightarrow \mathbb{P}^{rs+r+s}, ([a_0:...:a_r], [b_0,...,b_s]) \mapsto [a_0b_0,a_0b_1,...,a_0b_s,a_1b_0,...,a_rb_s]$.

What I am wondering is: if this is to be an embedding of anything besides sets, shouldn't I at least try to prove that this is a homeomorphism onto its image (where I am thinking of the domain in the product topology)? I guess what I'm asking is what sort of imbedding is this? I.e. what structure is being preserved?

Thanks for your time!

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    @M Turgeon: thanks. I've proved that fact for affine space. I guess the analogous statement for projective space is that the Segre embedding is NOT a homeomorphism onto its image when the domain has the product topology?2012-09-18

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