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 second page of the memorial article Claude Chevalley (1909–1984) by Dieudonné and Tits in Bulletin AMS 17 (1987) (doi:10.1090/S0273-0979-1987-15509-1), where they also say that Chevalley's introduction of the ideles was "a definite improvement on earlier similar ideas of Prüfer 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 Prüfer and von Neumann were doing, with references.]
I have two questions:
1) Can anyone point to a specific article where Prüfer 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.)