section filter
Let be a set and a non-empty net in . For each , define . Then the set
is a filter basis: is non-empty because , and for any , there is a such that and , so that .
Let be the family of all filters containing . is non-empty since the filter generated by is in . Order by inclusion so that is a poset. Any chain has an upper bound, namely,
By Zorn’s lemma, has a maximal element .
Definition. defined above is called the section filter of the net in .
Remark. A section filter is obviously a filter. The name “section” comes from the elements of , which are sometimes known as “sections” of the net .