Definition 3.4.2 (Generator and Subbase).label Let $X$ be a set, $\scr{O}\suf\cali{P}(X)$ and $\scr{F}$ a filter on $X$. We say $\scr{F}$ is the filter generated by $\scr{O}$ if $\scr{O}\suf\scr{F}$ is coarser than any filter $\scr{F}'$ that contains $\scr{O}$, and $\scr{O}$ is said to be a subbase of $\scr{F}$.

There exists a filter $\scr{F}$ containing $\scr{O}\suf \cali{P}(X)$ if and only if $\fall \curl{E_i}_{i\in I}\suf \scr{O}$ such that $I$ is finite, $\caps{}{i\in I}E_{i}\neq\emp$. In which case the filter generated by $\scr{O}$ has the form