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I was looking up a term called "derived net", however the Google search seems to conflate "net" with .NET programming language. (And filter with electronic filters, and "derived net from a filter" with implementation of Kalman Filter in VB.Net)

There is also a link to Modern General Topology By Jun-Iti Nagata, but I can't understand their notation at all.

Can someone please fill in the blank:

Let $(X, \mathfrak{F})$ be a topological space, and ($\mathcal{F}$, $\supseteq$) a direct set where $\mathcal{F}$ is a filter on $X$, the a net $\phi: \mathcal{F} \to X$ is a derived net of $\mathcal{F}$ if it has the property ___________.

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In these lecture notes (page 5) the condition seems simply to be that $\phi(A)\in A$ for every $A\in\mathcal F$.

As far as I can tell, this is also what Nagata defines -- except that both Nagata and the above notes use a separate directed set $\Delta$ (or $\mathscr D$) instead of $\mathcal F$ itself, but with an ordering given such that it is effectively just a relabeling of $\mathcal F$ anyway.

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  • $\begingroup$ And for many purposes the net generated by $\mathscr{F}$ is more useful: the directed set is $\mathscr{D}=\{\langle F,x\rangle:x\in F\in\mathscr{F}\}$, with $\langle F,x\rangle\ge\langle G,y\rangle$ iff $F\subseteq G$, and the net maps $\langle F,x\rangle$ to $x$. In a sense it encompasses all of the derived nets. $\endgroup$ – Brian M. Scott Jun 18 '16 at 21:17

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