TITLE: Brouwer's theorem for the Cauchy reals
QUESTION [17 upvotes]: 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.

REPLY [3 votes]: I received the following answer from an anonymous contributor:
Historically the first model with validity of the Brouwer 'theorem' was the topos $\mathrm{Sh}(N^N)$ of sheaves on Baire space. This goes apparently back to Dana Scott here:
Scott, Extending the topological interpretation to intuitionistic analysis II , pp.235-255 in Myhill,Kino,Vesley (eds.) Intuitionism and proof theory, North-Holland 1970. (pay-walled here)
The first installment of Scott is probably also of interest in the wider context and available from numdam
According to an exercise in Moerdijk- Mac Lane (p.345) $\mathrm{Sh}(N^N)$ has dependent choice as well, so it should fit your bill.
The question of validity of the Brouwer theorem was studied by Martin Hyland in the early seventies, relevant references are the following in the 'applications of sheaves' LNM, in particular the second:
Fourman, Hyland, Sheaf models of analyis, pp.280-301 LNM 753 Springer Heidelberg 1979. (PDF)
Hyland, Continuity in spatial toposes, pp.442-465 LNM 753 Springer Heidelberg 1979.  (pay-walled from springer page)
It seems the question then is, can $R_C$ satify the Brouwer thm with coinciding with $R_D$ ?