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Sep 5 at 21:32 comment added Manny Reyes Thanks for the tip about Fuchs's books, I will have to take a look!
Sep 5 at 3:25 comment added Pace Nielsen The finite topology is discussed quite a bit in L. Fuchs's books on (infinite) abelian groups. The concept was first developed (according to Fuchs) by Szele in the paper " On a topology in endomorphism rings of abelian groups". The topology is Hausdorff, linear, and (quite importantly) complete (in the net sense). In my own studies, the summability notion (of adding infinitely many endomorphisms) has been quite important.
Sep 4 at 18:52 vote accept Melanzio
Sep 4 at 17:07 history answered Manny Reyes CC BY-SA 4.0