Error in Lang's definition of weak topology?

Error in Lang's definition of weak topology?

On page 23-24 of his Real and Functional Analysis (3e) Serge Lang claims
Let $Y$ be a topological space and let $\mathscr{F}$ be a family of
mappings $f \colon X \to Y$ of $X$ into $Y$. Let $\mathscr{B}$ be the
family of all subsets of $X$ consisting of the sets $f^{-1}(W)$ where $W$
is open in $Y$ and $f$ ranges over $\mathscr{F}$. Then we leave to the
reader the verification of the following facts:
$\mathscr{B}$ is a base for a topology on $X$, i.e. satisfies conditions
B2 and B2.
...
Here B1 and B2 are given on p. 23
B1. Every element of $X$ lies in some set in $\mathscr{B}$.
B2. If $B$, $B'$ are in $\mathscr{B}$ and $x \in B \cap B'$ then there
exists some $B''$ in $\mathscr{B}$ such that $x \in B''$ and $B'' \subset
B \cap B'$.
It seems to me that the $\mathscr{B}$ defined in the first quote from Lang
need not satisfy the property B2 in the second quote.
For example, take $X=\Re^2$ , $Y=\Re$, $f(x_1,x_2) = x_1$,
$g(x_1,x_2)=x_2$, $\mathscr{F} = \{f,g\}$, $I=(0,1)$, $B=f^{-1}(I)$,
$B'=g^{-1}(I)$.
Then $B \cap B' = I \times I$ but no subset of this set can be an inverse
image under either $f$ or $g$ of any subset of $\Re$.
Am I right in believing that this is an error in the book?