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.

  • $\begingroup$ 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). $\endgroup$ Jul 31, 2017 at 5:07

4 Answers 4


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


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.


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).


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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