Talk:Adele ring

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Mathematics rating:	Start Class	Mid Priority	Field: Number theory
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Geometry guy 01:06, 15 June 2007 (UTC)

I don't understand the definition. What is the topology on adeles to make it a locally compact group? AFAICS the usual product topology is not locally compact in this case. EJ 14:51, 20 Aug 2004 (UTC)

Correct. But the restricted product topology is. The adeles as a whole are the union of more 'reasonable' subsets looking like product of p-adic integers over almost all p (compact), cartesian product with a finite product of p-adic numbers and the real numbers (locally compact). Every given adele is in such a 'reasonable' set.

Charles Matthews 16:24, 20 Aug 2004 (UTC)

Thanks for a reply. Notice that the restricted product page doesn't describe the topology either, which is why I asked in the first place. Just to make sure I understand you correctly: the res. prod. topology coincides with prod. topology on each 'reasonable' subset, thus a subset of the restricted product is open iff it is relatively open in each 'reasonable' subset; in other words, the res. prod. topology is inductively generated by prod. topologies on 'reasonable' subset. Is this correct?

EJ 11:16, 21 Aug 2004 (UTC)

Yes, that sounds right. Thanks for pointing out the deficiency of the RP page - should be fixed up. Charles Matthews 11:47, 21 Aug 2004 (UTC)