Editing Net (mathematics) (section)

==Relation to filters==
{{See also|Filters in topology#Filters and nets}}

A [[Filter (mathematics)|filter]] is a related idea in topology that allows for a general definition for convergence in general topological spaces. The two ideas are equivalent in the sense that they give the same concept of convergence.<ref>{{Cite web|url=http://www.math.wichita.edu/~pparker/classes/handout/netfilt.pdf|title=Archived copy|access-date=2013-01-15|archive-date=2015-04-24|archive-url=https://web.archive.org/web/20150424204738/http://www.math.wichita.edu/~pparker/classes/handout/netfilt.pdf|url-status=dead }}</ref> More specifically, every [[filter base]] induces an {{em|associated net}} using the filter's pointed sets, and convergence of the filter base implies convergence of the associated net. Similarly, any net <math>\left(x_a\right)_{a \in A}</math> in <math>X</math> induces a filter base of tails <math>\left\{\left\{x_a : a \in A, a_0 \leq a\right\} : a_0 \in A\right\}</math> where the filter in <math>X</math> generated by this filter base is called the net's {{em|eventuality filter}}. Convergence of the net implies convergence of the eventuality filter.<ref name="Bartle">R. G. Bartle, Nets and Filters in Topology, American Mathematical Monthly, Vol. 62, No. 8 (1955), pp. 551–557.</ref> This correspondence allows for any theorem that can be proven with one concept to be proven with the other.<ref name="Bartle" /> For instance, continuity of a function from one topological space to the other can be characterized either by the convergence of a net in the domain implying the convergence of the corresponding net in the codomain, or by the same statement with filter bases.

[[Robert G. Bartle]] argues that despite their equivalence, it is useful to have both concepts.<ref name="Bartle" /> He argues that nets are enough like sequences to make natural proofs and definitions in analogy to sequences, especially ones using sequential elements, such as is common in [[analysis]], while filters are most useful in [[algebraic topology]]. In any case, he shows how the two can be used in combination to prove various theorems in [[general topology]].

The learning curve for using nets is typically much less steep than that for filters, which is why many mathematicians, especially [[Mathematical analysis|analysts]], prefer them over filters. However, filters, and especially [[ultrafilter]]s, have some important technical advantages over nets that ultimately result in nets being encountered much less often than filters outside of the fields of analysis and topology.