Neighborhood base of weak topology

A neighbourhood basis of a point $x$ in a topological space is a family $\mathscr{B}_x$ of neighbourhoods of $x$ such that every neighbourhood of $x$ contains some member of $\mathscr{B}_x$. In particular, since the intersection of finitely many neighbourhoods of $x$ is again a neighbourhood of $x$, we must for every pair $V,W\in \mathscr{B}_x$ have an $U\in \mathscr{B}_x$ with $U \subset V\cap W$.

The intersection $N(f;\epsilon_1) \cap N(g;\epsilon_2)$ does in general not contain any set of the form $N(h;\epsilon_3)$, since for $f,g$ linearly independent, $N(f,\epsilon_1)\cap N(g;\epsilon_2)$ contains a linear subspace of codimension $2$, but no linear subspace of codimension $1$, while $N(h;\epsilon)$ contains a linear subspace of codimension $\leqslant 1$ (the kernel of $h$).

So we need to take finite intersections in order to have a neighbourhood-basis element contained in the intersection of two neighbourhood-basis elements.

The sets $N(f;\epsilon)$ form a neighbourhood-subbasis, however.