Brouwer famously proved, using principles motivated by intuitionistic choice sequences, that every function $\mathbb{R}\to \mathbb{R}$ is continuous. In *Sheaves in geometry and logic* (section VI.9), MacLane and Moerdijk exhibit a topos (the topos of sheaves on any sufficiently nice small full subcategory $\mathbf{T}\subseteq\mathrm{Top}$) in which this theorem holds in the internal logic. However, their proof is about the *Dedekind* real numbers. Can anyone point to a topos in which Brouwer's theorem holds for the *Cauchy* real numbers?


Of course it would suffice to give a topos in which Brouwer's theorem holds and in which the Cauchy and Dedekind real numbers coincide (e.g. if countable choice holds). This is not the case for MacLane and Moerdijk's example, in which $\mathbf{R}\_D$ is the sheaf of continuous $\mathbb{R}$-valued functions while (at least if all spaces in $\mathbf{T}$ are locally connected) $\mathbf{R}\_C$ is the sheaf of locally constant $\mathbb{R}$-valued functions. Moreover, it is easy to see that in this case there do exist discontinuous functions $\mathbf{R}\_C \to \mathbf{R}\_C$; so a different example is needed.



Brouwer's theorem for the Cauchy reals
Following Mike's suggestion, I post my comment as an answer.


Brouwer's theorem that all functions $\mathbb{R} \rightarrow \mathbb{R}$ are continuous holds in the effective topos. For example, this appears as Theorem 3.3.8 in van Oosten, *Realizability: An Introduction to its Categorical Side*. In fact it follows as a theorem from Markov's Principle and extended Church's thesis, which was apparently first discovered by Tsejtin.


