A subbase for the weak topology of real functions I'm reading a book chapter on weak topology, where it says that 

For a family $\mathscr{F}$ of real functions on $X$, the weak topology generated by $\mathscr{F}$ is denoted $\mathscr{\sigma}(X,\mathscr{F})$. It is easy to see that a subbase for $\mathscr{\sigma}(X,\mathscr{F})$ can be found by taking all sets of the form $U(f,x,\epsilon)=\{y\in{}X:|f(y)-f(x)|<\epsilon\}$.

Since the collection of all open intervals on $\mathbb{R}$ is a subbase for the topology on $\mathbb{R}$, I find it natural that the sets $\mathscr{B}=\{f^{-1}(I):I \text{ is an open interval in } \mathbb{R},\space{}f\in \mathscr{F}  \}$ form a subbase for $\mathscr{\sigma}(X,\mathscr{F})$. 
However, I don't see why $\tau(\mathscr{B})=\tau(U(f,x,\epsilon):f\in\mathscr{F},x\in{}X,\epsilon>0)$. Am I missing anything important here?

They are essentially the same. Their subbasis is the same as yours except that they only consider intervals in the image centered at some image point $f(x)$ and with diameter $2\epsilon$.
