It states in proposition 2 of GAGA that given an "algebraic space" (see FAC page 40) such as X= $P_r(\mathbb{C})$, we can take an its algebraic structure and view it as a unique analytic structure. In other words given chart $\phi: V_i \rightarrow U_i$ Z-open $V_i$ covering X, the chart can be viewed as an analytic chart from an open $V_i$ to analytic $U_i$ in $\mathbb{C}^n$.
where analytic means for every point x in $U_i$ there are functions $f_1...f_k$ holomorphic in a neighborhood W of x such that $U_i \cap W = \{z \mid f_i(z)=0\}$
My question is, is the unique topology of the "analytified" X simply the standard quotient topology with quotient map: $q:\mathbb{C}^n-\{0\} \rightarrow \mathbb{C}P^n$ see also:this post