Example

Let $\mathscr{S}$ be any set of subsets of a set $X$, and let $\mathfrak{T}$ be the topology on $X$ generated by $\mathscr{S}$ ($\S$ 2, no. 3, Example II). Since the set of finite intersections of sets of $\mathscr{S}$ is a base of $\mathfrak{T}$, it follows from the proof of Proposition 1 above and from Proposition 3 of $\S$ 1, no. 3 that for each $x \in X$ the neighbourhood filter of $x$ for $\mathfrak{T}$ is generated by the set $\mathfrak{G}(x)$ of sets of $\mathscr{S}$ which contain $x$.