Hartshorne, Corollary 2.3 question, which topology? I have a question about Corollary 2.3 (p. 11) of Hartshorne's algebraic geometry book.  Let
$$
D(x_i)= \{[z_0, \ldots, z_n]\in \mathbb{C}^{n+1} \mid z_i \neq 0\} \subset\mathbb{P}_n(\mathbb{C}) \, .
$$
Notice $\bigcup_{i=0}^n D(x_i) = \mathbb{P}_n(\mathbb{C})$.
Provided $Y$ a closed projective variety, he mentions a homeomorphism between (left) $Y \cap D(x_i)$ and (right) "an affine variety" $\phi_i |_Y(Y \cap D(x_i))$, where $\phi_i$ is a homeomorphism between D($x_i$) and $\mathbb{A}^n$.
My question is simple; what are the topologies for which we have a homeomorphism? He mentions an "induced topology". Can we just take the Zariski topology for both (right) and (left)?
See also this previous post.
 A: To understand the statement of the corollary, let's consider the the preceding proposition.  For $i = 0, 1, \ldots, n$, Hartshorne defines the map
\begin{align*}
\varphi_i : U_i &\to \mathbf{A}^n\\
[a_0 : a_1 : \cdots : a_n] &\mapsto \left(\frac{a_0}{a_i}, \ldots, \widehat{\frac{a_i}{a_i}}, \ldots, \frac{a_n}{a_i}  \right)
\end{align*}
where the hat indicates that $a_i/a_i$ is omitted.  (Here $U_i = D(x_i)$.)  $\mathbf{P}^n$ is given the usual (homogeneous) Zariski topology, and $U_i \subseteq \mathbf{P}^n$ is given what Hartshorne calls the induced topology, but what I like to call the subspace or relative topology: we define the open sets of $U_i$ to be those of the form $V \cap U_i$ where $V$ is an open subset of $\mathbf{P}^n$.
Proposition 2.2. The map $\varphi_i$ is a homeomorphism of $U_i$ with its induced topology to $\mathbf{A}^n$ with its Zariski topology.
Corollary 2.3 follows easily from this proposition.  Given a closed projective variety $Y \subseteq \mathbf{P}^n$, consider the restrictions of the above maps $\varphi_i|_Y: Y \to \mathbf{A}^n$.  In the corollary, Hartshorne claims that $\varphi_i|_Y(Y \cap U_i) \subseteq \mathbf{A}^n$ is an affine variety (again, equipped with the subspace topology inherited from $\mathbf{A}^n$) and
$$
\varphi_i: Y \cap U_i \overset{\sim}{\longrightarrow} \varphi_i|_Y(Y \cap U_i)
$$
is a homeomorphism.
