Supernatural numbers and a new topology on the arithmetic site
  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 $\mathbb{Q}^*_+ \backslash \mathbb{A}^f_{\mathbb{Q}} / \widehat{\mathbb{Z}}^*$. The induced topology of $\mathbb{A}^f_{\mathbb{Q}}$ 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 $\overline{\mathbf{Spec}(\mathbb{Z})}/\mathbb{F}_1$: it is compact, has an uncountable basis of opens, each non-empty open being dense, and it satisfies the $T_1$ separation property for incomparable points. 
Lieven Le Bruyn
math.RA