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What are good resources (books, lecture notes, introductory papers, ...) to learn about convergence spaces?

I read in an answer to a recent question the following:

There's still a subtle issue. If you try to axiomatize topological spaces via filter convergence, you wind up with a significantly more general notion called a convergence space. Convergence spaces are really nice, and personally I wish people would start treating them as the basic objects of interest in general topology, and treat topological spaces as a mere special case. Unfortunately, I haven't been able to find an elementary introduction to such things that I can link you to, so you'll have to learn things the classical way until you're ready to go off on your own.

This sounds quite intriguing, so I thought it might be reasonable to ask where to look if I decide to learn a bit more about convergence spaces.

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  • $\begingroup$ I was not sure whether to add the tag (category-theory) or not - I added it mainly because most often when I see convergence spaces, they are introduced as a category (and quite often they appear in texts about category theory). $\endgroup$
    – Martin Sleziak
    Jul 31, 2017 at 5:07

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There is a new (2016) textbook which covers convergence spaces and related concepts:

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Some suggestions:

and references therein. I'm also quite interested to learn about new resources, especially ones with lots of (worked) examples to get a feel for the difference between, say, a pseudotopological and a pretopological convergence, etc.

The motivation for their study often seems category-theoretical (convergence spaces are Cartesian closed, e.g.). I would also like to see a good list of open problems, e.g.

This page offers a general overview of "topology-like" classes of spaces, and I once read parts of the book: "G. Preuss, "Topological structures—An approach to categorical topology" , Reidel (1988)", which mostly talks about category constructions (like initial and final structures), it seems appropriate in view of your tags.

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Additionally, there is:

  • "Problems in the Theory of Convergence Spaces" by D. R. Patten
  • "Grundstrukturen der Analysis" I & II by W. Gähler, if you understand German

Also "Handbook of Analysis and its Foundations" by E. Schechter deals briefly with convergence spaces (and pretopological spaces).

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I think the book below is trying to do exactly what you are looking for:

Szymon Dolecki and Frédéric Mynard, Convergence Foundations of Topology. World Scientific Publishing. 2016. DOI:10.1142/9012

There's also this upcoming one, somewhat in the same spirit:

Szymon Dolecki, A Royal Road to Topology. World Scientific Publishing. 2023. DOI:10.1142/12154

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The one I'm reading is Convergence Structures and Applications to Functional Analysis by R. Beattie and Heinz-Peter Butzmann.

I've liked it so far, and starts with Convergence Spaces right away. Worth taking a look, specially for the ones that like Analysis.

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