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In Hartshorne's "Algebraic Geometry" p. 77, Example 2.5.1, it is mentioned that if "$k$ is an algebraically closed field, then the subspace of closed points of $\operatorname{Proj} \, k[x_0,\cdots,x_n]$ is naturally homeomorphic to the projective $n$-space $\mathbb{P}^n$. He refers to Ex. 2.14d, however I don't see the connection. Any insights?

Thanks.

P.S. Ex. 2.14(d) seems to me a little bit obscure at this point, this is why i am not reproducing it. Any argument relating $\operatorname{Proj} \, k[x_0,\cdots,x_n]$ and $\mathbb{P}^n$ is very welcome.

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    @MTurgeon: Thanks. I've been studying proposition 2.6 since the day before yesterday, it takes some time to digest :)2012-10-05

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Exercise $2.14~ d)$ states that for any projective variety with homogeneous coordinate ring $S$, $t(V) \simeq \operatorname{Proj}S$ Which include $\mathbb{P}^{n}$, meaning $V$ could be $\mathbb{P}^{n}$.

Now by proposition 2.6, $V$ and $t(V)$ have homeomorphic closed points.

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    Sorry, I had gone away from the computer and missed the follow up comments. Glad someone else helped. =)2012-10-07