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From Planetmath:

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 $$S:=\lbrace S(j)\mid j\in D\rbrace$$ 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, $$\mathcal{F}:=\bigcup_{i=1}^{\infty} \mathcal{F}_i.$$ By Zorn's lemma, $\mathcal{A}$ has a maximal element $\mathcal{X}$. $\mathcal{X}$ defined above is called the section filter of the net $(x_i)$ in $X$.

  1. I was wondering if the sentence in bold is contrary to the definition of $S(j)$? Under that definition, it should be $S(\ell) \supseteq S(j)\cap S(k)$ instead, which means $S$ cannot be a filter basis.
  2. Isn't the maximal element $\mathcal{X}$ always the power set of $X$?
  3. How is the "section filter" of a net related to the filter generated by the net? The filter generated by a net $(x_i)_{i\in D}$ that I found out elsewhere in the internet is defined as

    For each $j \in D$, define $x_j := \{x_i : i ≥ j\}$. The collection of tails $\{x_j : j ∈ D\}$ is a filterbase on $X$. The filter generated by this filterbase is called the filter generated by the net.

Thanks and regards!

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1 Answer 1

up vote 2 down vote accepted

The problem is that $S(j)$ has been defined incorrectly: the definition should read $$S(j):=\{x_i:j\le i\}\;.$$ In other words, you want to look at the family of tails. It’s probably just a typo, either for $\{x_i:j\le i\}$ or for $\{x_i:i\ge j\}$. In short, it is the filter generated by the net in the sense of the other definition that you quoted.

Note that $\mathcal{X}$ can’t be $\wp(X)$, since $\mathcal{X}$, being a filter, can’t contain $\varnothing$. More generally, it can’t contain any set disjoint from an $S(j)$, since it definitely contains every $S(j)$.

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I don't think I've seen anyone use \wp for power set before. Isn't it usually only used for Weierstrass' elliptic functions? –  kahen Feb 3 '12 at 21:50
    
@kahen: I prefer to reserve $\mathscr{P}$ for other purposes, so I use $\wp$ for the power set. –  Brian M. Scott Feb 3 '12 at 22:18
    
Well... there's also $\mathcal P$ (\mathcal P) –  kahen Feb 3 '12 at 23:20
    
@kahen: Which is just plain ugly. :-) I dislike the mathcal font, and $\wp$ actually looks a lot like my handwritten power set symbol. –  Brian M. Scott Feb 3 '12 at 23:26
    
@BrianM.Scott: Thanks! (1) "it is the filter generated by the net in the sense of the other definition that you quoted", do you mean $S$ is the filter base generated by the net? (2) Is the filter generated by a net the minimal one containing the filter base generated by the net? (3) Is the section filter $\mathcal{X}$ of the net $(x_i)$ in $X$ usually not the same as the filter generated by the net? –  Tim Feb 20 '12 at 13:16

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