filter basis

A filter subbasis for a set $S$ is a collection of subsets of $S$ which has the finite intersection property.

A filter basis $B$ for a set $S$ is a non-empty collection of subsets of $S$ which does not contain the empty set such that, for every $u\in B$ and every $v\in B$, there exists a $w\in B$ such that $w\subset u\cap v$.

Given a filter basis $B$ for a set $S$, the set of all supersets of elements of $B$ forms a filter on the set $S$. This filter is known as the filter generated by the basis.

Given a filter subbasis $B$ for a set $S$, the set of all supersets of finite intersections of elements of $B$ is a filter. This filter is known as the filter generated by the subbasis.

Two filter bases are said to be equivalent if they generate the same filter. Likewise, two filter subbases are said to be equivalent if they generate the same filter.

Note: Not every author requires that filters do not contain the empty set.Because every filter is a filter basis then accordingly some authors allow that a filterbase can contain the empty set.