Corollary

Two filter bases $\mathfrak{B}, \mathfrak{B}'$ on a set $X$ are equivalent if and only if every set of $\mathfrak{B}$ contains a set of $\mathfrak{B}'$ and every set of $\mathfrak{B}'$ contains a set of $\mathfrak{B}$.

Examples of filter bases. 1) Let $X$ be a topological space. Proposition 3 shows that the bases of the neighbourhood filter of a point $x \in X$ are precisely the fundamental systems of neighbourhoods of $x$ (§ 1, no. 3, Definition 5).

2) Let $X$ be a non-empty directed set with respect to a relation $(\sigma)$ (Set Theory, Chapter III, § 1, no. 10). For each $a \in X$, the set $S(a)$ of all $x \in X$ such that $a(\sigma)x$ will be called the section of $X$ relative to the element $a$. Then the set $\mathscr{S}$ of sections of $X$ is a filter base, for it clearly satisfies $(\mathrm{B}_{\mathrm{II}})$, and if $a, b$ are any two elements of $X$, then there is by hypothesis an element $c \in X$ such that $a(\sigma)c$ and $b(\sigma)c$, and therefore

so that $(\mathrm{B}_1)$ is satisfied. The filter generated by $\mathscr{S}$ is called the section filter of the directed set $X$.

For example, the Fréchet filter (no. 1) is the section filter of the ordered set $\mathbf{N}$, considered as directed by the relation $\leqslant$. Let $\mathfrak{F}$ be a filter on a set $Z$. Since $\mathfrak{F}$ is directed with respect to the relation $\supset$ [by reason of axiom $(\mathrm{F}_{\mathrm{II}})$] we can define a section filter on $\mathfrak{F}$; here a section of $\mathfrak{F}$ relative to a set $A \in \mathfrak{F}$ is the set $S(A)$ of all $M \in \mathfrak{F}$ such that $M \subset A$. This filter is called the section filter of the filter $\mathfrak{F}$.