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.
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.