section filter

Let $X$ be a set and $(x_{i})_{i\in D}$ a non-empty net in $X$ . For each $j\in D$ , define $S(j):=\{x_{i}\mid i\leq j\}$ . Then the set

S:=\{S(j)\mid j\in D\}

is a filter basis: $S$ is non-empty because $(x_{i})\neq\varnothing$ , and for any $j,k\in D$ , there is a $\ell$ such that $j\leq\ell$ and $k\leq\ell$ , so that $S(\ell)\subseteq S(j)\cap S(k)$ .

Let $\mathcal{A}$ be the family of all filters containing $S$ . $\mathcal{A}$ is non-empty since the filter generated by $S$ is in $\mathcal{A}$ . Order $\mathcal{A}$ by inclusion so that $\mathcal{A}$ is a poset. Any chain $\mathcal{F}_{1}\subseteq\mathcal{F}_{2}\subseteq\cdots$ has an upper bound, namely,

\mathcal{F}:=\bigcup_{i=1}^{\infty}\mathcal{F}_{i}.

By Zorn’s lemma, $\mathcal{A}$ has a maximal element $\mathcal{X}$ .

Definition. $\mathcal{X}$ defined above is called the section filter of the net $(x_{i})$ in $X$ .

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

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