The ur-solenoid and the adeles

Most of the discussion of the 2-solenoid did not depend upon the 2, apart from the fact that 2 is an integer larger than 1. We consider a larger family of solenoids, at first with 2 replaced by a larger integer N , suggesting that such limits factor according to the prime factorization of N . This factorization does occur, and requires a proof which can be given in a general context. Incidentally, to make the universal ur-solenoid, we must also improve our indexing of limits, and understand the effect of variation of indexing on the limit object. Finally, we do make the ur-solenoid (in notation below)