Definition 3

A set $\mathfrak{B}$ of subsets of a set $X$ which satisfies axioms (B_I) and (B_{II}) is said to be a base of the filter it generates. Two filter bases are said to be equivalent if they generate the same filter.

If $\mathfrak{G}$ is a subbase of a filter $\mathfrak{F}$, then the set $\mathfrak{G}'$ of finite intersections of sets of $\mathfrak{G}$ is a base of $\mathfrak{F}$ (no. 2).