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# 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 filter base can contain the empty set.

## Mathematics Subject Classification

03E99*no label found*54A99

*no label found*

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