The ever-reliable Wikipedia [says](http://en.wikipedia.org/wiki/Idele):

>... an adelic algebraic group is a semitopological group defined by...

No more details are given, and I was wondering if the multiplication only being _separately_ continuous has any noticeable effect when working with adelic groups. I ask because there are some algebras which pop up in the study of von Neumann algebras where the multiplication is not jointly continuous, and this is mildly annoying.