When reading older papers, I often see references to bornological or uniform spaces, which encode the notions of "boundedness" or "uniformness". In this way, they seem to sit between topological spaces and metric spaces.
It seems like these used to be of some amount of interest (they are explicitly mentioned in Rudin's "Functional Analysis" as topics he's omitting) so it seems odd to me that I've never heard them mentioned in a class, talk, etc. It might just be uncommon in my department, but I don't think I've ever seen them mentioned on MSE either, so they must be somewhat uncommon.
Looking at the axioms, they certainly look rather unwieldy, but you could say the same about lots of other mathematical objects which are quite well studied.
What, then, happened to bornological spaces and uniform spaces?
Does anyone still do work in this area? Has it been subsumed by some other topic? This mystery haunts me.
Thanks in advance! ^_^
When I first read your question, it seemed to me that uniform spaces were much more used than bornological spaces. But as I frequently use uniform spaces in my own research, I thought that my opinion might be biased and I looked for more measurable criteria.
(1) It turns out that both topics have their specific entry in the MSC2020-Mathematics Subject Classification System: 46A08 Barrelled spaces, bornological spaces, 54E15 Uniform structures and generalizations. A rough research on MathSciNet for these codes (since 2000) gives 43 entries for 46A08 and 229 entries for 54E15.
(2) A keyword research on MSE gives 316 results for "uniform space" but only 8 results for "bornological space". (the "_" are important: if you omit them and just look for uniform space, you get 8,297 results)
(3) The same research on Mathoverflow gives 98 results for "uniform space" but only 6 results for "bornological space".
Also note there is a tag uniform-spaces on both MSE and Mathoverflow but no tag for "bornological-spaces". All of this seem to confirm my original feeling.