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I have touched net, just adding some more hyperlinks and cross-references within the page. Also filter, where I made eventuality filter come out as a link.
I have been expanding at net a fair bit, also reformatting (more numbered environments) and re-arranging (more subsections, the order of some paragraphs reversed, where it seemed to help the reader).
Mainly I wanted to, and did, add statement and detailed proof of these four facts:
(The last of these statements I also copied to a stand-alone entry compact spaces equivalently have converging subnet of every net. )
Most of the other edits to the entry that I did were meant to make the statement of these proofs flow nicely. To that end for example I made explicit the classes of examples of directed sets that are needed in the proofs, in a Backgrounds-subsection “Directed sets”.
And since I added all these proofs in the subsection “Relation to topology” (now “Properties – Relation to topology”) I moved the plain definition of convergence of nets in topological spaces, which used to be there, to the section Definition – Nets.
At other places I simply expanded out the originally somewhat terse discussion to (hopefully) more easily readible text, such as in the section Nets and filters.
Finally, I considerably expanded the Idea-section.
All in all, while I did edit a lot, I tried to retain everything that used to be there, if maybe slightly re-arranged. But Toby should please check if he can live with the edits.
The correspondence betwen nets and filters seems very reminiscent of the correspondence between fibrations and presheaves, especially the construction of a filter net as a directed system of pointed sets seems very similar to the Grothendieck construction
Is there any way to make this analogy more precise?
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