TITLE: Relation between $\operatorname{Proj} \, k[x_0,\cdots,x_n]$ and $\mathbb{P}^n$
QUESTION [6 upvotes]: 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.

REPLY [6 votes]: 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.