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[parent] section filter (Definition)

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):=\lbrace x_i\mid i\le j\rbrace$. Then the set

$\displaystyle S:=\lbrace S(j)\mid j\in D\rbrace$
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\le \ell$ and $ k\le \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,

$\displaystyle \mathcal{F}:=\bigcup_{i=1}^{\infty} \mathcal{F}_i.$
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)$.



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Cross-references: maximal element, Zorn's lemma, upper bound, chain, poset, inclusion, order, filter generated by, filters, filter basis, net
There are 2 references to this entry.

This is version 1 of section filter, born on 2007-02-12.
Object id is 8906, canonical name is SectionFilter.
Accessed 725 times total.

Classification:
AMS MSC: 54A99 (General topology :: Generalities :: Miscellaneous)
 03E99 (Mathematical logic and foundations :: Set theory :: Miscellaneous)
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