Supernatural numbers and a new topology on the arithmetic siteSupernatural numbers and a new topology on the arithmetic site
Faculty of Sciences. Mathematics and Computer Science
(2014), p. 1-9
University of Antwerp
In arXiv:1405.4527 Connes and Consani introduced and studied the arithmetic site and showed that the isomorphism classes of points are in canonical bijection with the finite adele classes Q∗+∖AfQ/Z∗. The induced topology of AfQ on this set is trivial, whence this space is usually studied via noncommutative geometry. However, we can define another topology on this set of points, which shares several properties one might expect of the mythical object Spec(Z)¯¯¯¯¯¯¯¯¯¯¯¯/F1: it is compact, has an uncountable basis of opens, each non-empty open being dense, and it satisfies the T1 separation property for incomparable points.