One of the best-known results in mathematics is the uncountability of the real numbers, which Georg Cantor proved by the diagonalization method. His proof relies on the law of excluded middle or the axiom of choice. In joint work with James E. Hanson we showed that this is necessary by constructing a mathematical universe, the topos of countable reals, in which the Dedekind reals are countable, and consequently both the axiom of choice and the law of excluded middle are invalid. The construction rests on a piece of classical topology, a generalization of Kakutani’s fixed-point theorem.

In this talk we shall explore how topology behaves in the topos of countable reals. In many respects the reals are still well behaved. They form a Dedekind-complete archimedean ordered field and are connected. The closed interval is totally bounded and Cauchy-complete, although it lacks the stronger Heine–Borel property, as it can be covered by intervals whose lengths sum up to any desired small positive real. Brouwer’s fixed-point theorem holds, as a corollary of the countability of the Hilbert cube and Lawvere’s fixed-point theorem. Whether every function on the reals is continuous remains open.