section filter

Let X be a set and (xi)iD a non-empty net in X. For each jD, define S(j):={xiij}. Then the set


is a filter basis: S is non-empty because (xi), and for any j,kD, there is a such that j and k, so that S()S(j)S(k).

Let 𝒜 be the family of all filters containing S. 𝒜 is non-empty since the filter generated by S is in 𝒜. Order 𝒜 by inclusion so that 𝒜 is a poset. Any chain 12 has an upper bound, namely,


By Zorn’s lemma, 𝒜 has a maximal elementMathworldPlanetmath 𝒳.

Definition. 𝒳 defined above is called the section filter of the net (xi) in X.

Remark. A section filter is obviously a filter. The name “sectionPlanetmathPlanetmath” comes from the elements S(j) of S, which are sometimes known as “sections” of the net (xi).

Title section filter
Canonical name SectionFilter
Date of creation 2013-03-22 16:41:37
Last modified on 2013-03-22 16:41:37
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 4
Author CWoo (3771)
Entry type Definition
Classification msc 54A99
Classification msc 03E99