Proposition 8

Let $\mathfrak{F}$ be a filter on a set $X$ and $A$ a subset of $X$. Then the trace $\mathfrak{F}_A$ of $\mathfrak{F}$ on $A$ is a filter if and only if each set of $\mathfrak{F}$ meets $A$.

Since $(M \cap N) \cap A = (M \cap A) \cap (N \cap A)$ we see that $\mathfrak{F}_A$ satisfies $(\mathrm{F}_{\mathrm{II}})$; again, if $M \cap A \subset P \subset A$ then $P = (M \cup P) \cap A$, whence $\mathfrak{F}_A$ satisfies $(\mathrm{F}_1)$. Hence $\mathfrak{F}_A$ is a filter if and only if it satisfies $(\mathrm{F}_{\mathrm{III}})$, i.e. if and only if each set of $\mathfrak{F}$ meets $A$.

In particular, if $A \in \mathfrak{F}$ then $\mathfrak{F}_A$ is a filter on $A$, by $(\mathrm{F}_{\mathrm{II}})$ and $(\mathrm{F}_{\mathrm{III}})$.