Initial topology I have read that for all $U \in \sigma (X, {f_i, i \in I})$ (initial topology) there exists a finite number of open sets $V_i$ in $T_{Y_i}$ s.t. 
$U = \bigcap_{i\in G}f_i^{-1}(V_i)$ or $U = \bigcup_{i\in J}f_i^{-1}(V_i).$
I doubt that all. As I know $f_i$ may not be a bijection. So, we can have $U$ s.t. no $f_i^{-1}$ map onto it. Where is a mistake?

The family of finite intersections of $f_i^{-1}(V_i)$ form a basis of the topology, but in general, there are open sets that are neither a finite intersection of sets $f_i^{-1}(V_i)$ nor a union of $f_i^{-1}(V_i)$.
Consider the product topology on $\mathbb{R}^2$, which is the initial topology with respect to the coordinate projections.
The sets of the form $\bigcap_{i\in G} f_i^{-1}(V_i)$ are the open rectangles $U_1 \times U_2$, and the sets of the form $\bigcap_{i\in I} f_i^{-1}(V_i)$ are unions of strips $U_1 \times \mathbb{R} \cup \mathbb{R}\times U_2$. A ball in the Euclidean metric is not of either form.
