I am teaching a course leading up to Tate's thesis and I told the students last week, when defining ideles, that the first topology that was put on the ideles was not so good (e.g., it was not Hausdorff; it's basically the profinite topology on the ideles, so archimedean components don't get separated well). You can find this mentioned on the 2nd page of the memorial article on Chevalley by Dieudonne and Tits in AMS Bulletin 17 (1987) (read it here:
http://www.ams.org/journals/bull/1987-17-01/S0273-0979-1987-15509-1/S0273-0979-1987-15509-1.pdf), where they also say that Chevalley's introduction of the ideles was "a definite improvement on earlier similar ideas of Prufer and von Neumann, who had only embedded *K* [the number field] into the product over the *finite* places" (emphasis theirs). [Edit: Scholl's answer says in a little more detail what Prufer and von Neumann were doing, with references.]

I have two questions:

1) Can anyone point to a specific article where Prufer or von Neumann used a product over just the finite places, or at least indicate whether they were able to do anything with it?

2) Who introduced the restricted product topology on the ideles? (In Chevalley's 1940 paper deriving global class field theory using the ideles and not using complex analysis, Chevalley uses a different topology, as I mentioned above.) I would've guessed it was Weil, but BCnrd told me that he heard it was due to von Neumann. Any answer with some kind of evidence for it is appreciated.

Edit: For those wondering why the usual notation for the ideles is J_K and not I_K, the use of J_K goes right back to Chevalley's papers introducing ideles. (One may imagine I_K could have been taken already for something related to ideals, but in any event it's worth noting the use of "J" wasn't some later development in the subject.)

le groupe des idèles de$k$; c'est un groupe abélien séparé, localement compact. – Chandan Singh Dalawat Oct 6 '10 at 12:37