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Nets are an important generalization of sequences. Equivalently, convergence of sequences and the like can be formulated using filters instead of nets.

As for the notion of sub-sequence, it can be expressed in terms of nets by taking a cofinal subset of the directed indexing set. But how do you express sub-sequences in terms of filters?

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    $\begingroup$ Are you looking for the definition of a subfilter of a filter $\mathcal{F}$? This is really just a filter $\mathcal{G}$ with $\mathcal{G} \subseteq \mathcal{F}$. So the definition is much easier than for subnets. Or do you want to know if some theorems about subnets (which ones?) carry over to subfilters? $\endgroup$ – HeinrichD Oct 2 '16 at 7:00
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    $\begingroup$ Cofinal subsets of the directed indexing set are not the "right" notion of subnet. If, for example, in the characterization of compactness we want every net to have a convergent subnet, then we should allow cofinal but not necessarily injective maps out of other directed sets. Say, every sequence has a convergent subnet in this sense, but not necessarily a convergent subsequence. $\endgroup$ – Alexander Shamov Oct 2 '16 at 14:06
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    $\begingroup$ @HeinrichD As explained in Michael Greinecker's answer, the inclusion of filters in your comment should be the other way around. Subnets correspond to larger filters. $\endgroup$ – Andreas Blass Oct 2 '16 at 14:09
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    $\begingroup$ I heartily recommend this PDF, ‘Translating Between Nets and Filters’, by Saitulaa Naranong. $\endgroup$ – Brian M. Scott Oct 2 '16 at 21:54
  • $\begingroup$ In connection with this it is worth mentioning that several different definition of subnets are used. In the case of AA-subnets, relation between subnets and finer filters is rather trivial. Some helpful answers and links can be found here and here. $\endgroup$ – Martin Sleziak Nov 18 '16 at 9:15
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For every net $\langle x_\alpha\rangle_{\alpha\in A}$, you call a set of the form $\{x_\beta : \beta\in A, \;\beta\succeq\alpha\}$ a tail-set. The tail-sets form a filter base, and the filter generated by the tail-sets is the eventuality filter of the net. The eventuality filter of a subnet will be larger under set inclusion then. The characterization of compactness that every net has a convergent subnet then turns into the corresponding argument that for every filter there exists a larger filter that converges. Equivalently, maximal filters,always converge and these are just the ultrafilters.

To get an actual correspondence, we have to show that every filter is the eventuality filter of some net. Take a filter $\mathcal{F}$. Ordering $X\times\mathcal{F}$ by $(x,B)\succeq(y,B')$ iff $B\subseteq B'$ we get a directed set and with this directed set, $(x,B)\mapsto x$ becomes a net with eventuality filter $\mathcal{F}$.

A great source for material on the relationship between nets and filters is the Handbook of Analysis and its Foundations by Eric Schechter.

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