Subbasis of a weak topology Let $\{(X_\alpha,\mathscr{T}_\alpha):\alpha\in\Lambda\}$ be an indexed family of topological spaces, and for each $\alpha\in\Lambda$ let $f_\alpha:X\to X_\alpha$ be a function. Furthermore, let $\mathscr{T}$ be the weak topology on X induced by $\{f_\alpha:\alpha\in\Lambda\}$. Then $\mathscr{L}=\{f_\alpha^{-1}(U_\alpha):\alpha\in\Lambda; U_\alpha\in\mathscr{T}_\alpha\}$ is a subbasis for $\mathscr{T}$.
I have tried this: Since $f_\alpha$ is continuous therefore $f_\alpha^{-1}(U_\alpha)$ is open then, intersections of elements of $\mathscr{L}$ are open. Let $\mathscr{B}$ be the set of all of those finite intersections then if $B_1,B_2\in\mathscr{B}$ and $x\in B_1\cap B_2$, ($x\in X$), then there exist $B\in\mathscr{B}$ such that $x\in B$ and $B\subseteq B_1\cap B_2$. I need to prove that $X=\cup\{B:B\in\mathscr{B}\}$.


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*First I would like to know some important properties of a weak topology.

*And of course I would like to see a proof of this theorem please.

 A: It looks to me as if what you are calling "weak topology" in your question is commonly referred to as initial topology. 
As for the question why the set you describe is a subbase for the weak topology on $X$: I am not sure there is anything to show. That it is a subbase means that it generates the (weak) topology that is, that the weak topology is the smallest topology containing $\mathscr L$ as a subset. But if every map $f_\alpha$ is to be continuous, the topology must contain at least all sets in $\mathscr L$. But the weak topology is defined to be the smallest topology such that all $f_\alpha$ are continuous. Hence they have to be equal.
A: For a set $X$ with maps $f_\alpha: X \rightarrow X_\alpha$, where $X_\alpha$ are topological spaces, the weak topology is the smallest topology on $X$ such that the $f_\alpha$ are continuous (explicitly, one can take it to be the intersection of all topologies on $X$ for which $f_\alpha$ are continuous).
To prove your claim, just show that $\mathcal{L}$ have to be open in any topology for which $f_\alpha$ are continuous. 
