I have a some troubles with the next exercise.
Let X be a topological spaces and $\{X_j:j\in J\}$ be a family of topological spaces. For all $j\in J$ let $f_j:X\rightarrow X_j$ be a continuous function. We say that the family $\mathcal{C}=\{f_j:j\in J\}$ separates points of closed sets in $X$ if for all $F\subseteq X$ closed set and for all $x\in X\setminus F$, exist $j\in J$ such that $f_j(x)\notin\text{cl}_{X_j}(f_j[F])$ where $\text{cl}_{X_j}$ is the closure of a set in the $X_j$ space.
1) Prove that $\mathcal{C}$ separates points of closed sets in $X$ if and only if the family $A=\{f_j^{-1}[U]|j\in J, U\in\tau_j\}$ is a basis for $X$
2) Prove that if $\mathcal{C}$ separates points of closed sets in $X$, then $X$ has the weak topology induced by $\mathcal{C}$
My attempt:
1)
$\Rightarrow)$
Clearly, $\bigcup A=X$, Then, if we consider $B,C\in A$, we need find some $D\in A$ such that $D\subseteq B\cap C$
Let $f^{-1}_j[Uj],f^{-1}_i[U_i]\in A$. Then, if $x\in f^{-1}_j[Uj]\cap f^{-1}_i[U_i]$ , so, $x\notin X\setminus(f^{-1}_j[Uj]\cap f^{-1}_i[U_i])$ and clearly this set is closed, because $f^{-1}_j[Uj]\cap f^{-1}_i[U_i]$ is open because $f_{i,j}$ is continuous and $U_{i,j}$ is open. Then, by hypothesis there exist $k\in J$ such that $$f_k(x)\notin \text{cl}_{X_k}\left(f_k\left[X\setminus(f^{-1}_j[Uj]\cap f^{-1}_i[U_i]) \right] \right)$$Then, $f_k(x)\in X_k\setminus \text{cl}_{X_k}\left(f_k\left[X\setminus(f^{-1}_j[Uj]\cap f^{-1}_i[U_i]) \right] \right)$
In this way, $x\in f^{-1}_k\left[ X_k\setminus \text{cl}_{X_k}\left(f_k\left[X\setminus(f^{-1}_j[Uj]\cap f^{-1}_i[U_i]) \right] \right)\right]=X\setminus\text{cl}_{X_k}(f_k[X\setminus f^{-1}_j[U_j]\cap f^{1}_i[U_i]])$ and this set is open, again, because $f$ is continuos. I think that this set can be $D$, but, how can I prove it?
$\Leftarrow)$
Let $F\subseteq X$ be a closed set and $x\in X\setminus F$. We know that $X\setminus F$ is an open set, then, like $A$ is a basis, $X\setminus F=\displaystyle\bigcup_{i\in I}^{}f^{-1}_i[U]$.
Like $x\in X\setminus F$, then, there exist $i\in I$ such that $x\in f^{-1}_i[U]$ and $f^{-1}[U]$ is open. Then, $f_i(x)\in U$ and $U\subseteq f_i[X\setminus F]$. From here, I don't know how can I conclude.
2)
If we take another topology $\tau$ such that for all $i\in J$ $f_i$ is continuos, then, the topology generated by $A$, denoted by $\tau_A$ holds $\tau_A\subseteq\tau$. Then, $\tau_A$ is the weak topology. This fact is because if $\mathcal{C}$ separetes points of closed sets, then, $A$ is a basis for some topology.
I really appreciate any help you can provide me.