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Let $Y\subseteq \mathbb{A}^n$ be an affine variety, $\bar{Y}$ be its projective closure in $\mathbb{P}^n$. In GTM 52 of Robin Hartshone, there is a problem of finding generator of $I(\bar{Y})$(problem 2.9)

Suppose that $\varphi : \mathbb{A}^n \rightarrow U_0 $ be the homeomorphism between two spaces, then is $\varphi(Y)$ is the projective closure of $Y$ ?

How can I imagine about the projective closure of an affine variety, topologically ?

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