  Ali Enayat had asked whether there is a nonstandard model of Peano arithmetic
(PA) that can be represented as $\langle\mathbb{Q},\oplus,\otimes\rangle$,
where $\oplus$ and $\otimes$ are continuous functions on the rationals
$\mathbb{Q}$. We prove, affirmatively, that indeed every countable model of PA
has such a continuous presentation on the rationals. More generally, we
investigate the topological spaces that arise as such topological models of
arithmetic. The reals $\mathbb{R}$, the reals in any finite dimension
$\mathbb{R}^n$, the long line and the Cantor space do not, and neither does any
Suslin line; many other spaces do; the status of the Baire space is open.
