Definition 5

Let $A$ be a subset of a set $X$ and $\mathfrak{F}$ a filter on $X$. If the trace of $\mathfrak{F}$ on $A$ is a filter on $A$, this filter is said to be induced by $\mathfrak{F}$ on $A$.

If a filter $\mathfrak{F}$ on $X$ induces a filter on $A \subset X$, then the trace on $A$ of a base of $\mathfrak{F}$ is a base of $\mathfrak{F}_A$, by reason of Proposition 3 of no. 3.