The classical computability theory taking place in $\mathbb{N}$, can be extended to more general spaces, like $T\_0$ second countable topological spaces $(X, \mathcal{O}, v)$ where $\mathcal{O}$ is a countable basis of $X$ and $v:\mathbb{N} \rightarrow \mathcal{O}$ a total surjection. Then we say that $f:X \rightarrow Y$ is computable if given any enumeration of all basic open sets containing $x$, one can compute (in the sense of Turing) an enumeration of all basic open sets contaning $f(x)$. Equivalently we can state that for any basic open set $O$ of $Y$, we have that $f^{-1}(O)$ is a recursively enumerable open in the sence that $f^{-1}(O)=\cup\_n O\_n$ where $O\_n$ is a recursively enumerable sequence of basic open sets of $X$


This brings some sort of a generalization of Turing reduction, when $X$ and $Y$ are both the Cantor space.


Now, I am wondering if it is possible to have a categorical point of view on all this. My research brought me to Topoi, the effective topos and the recursive topos. I have some difficulties to understand what is the effective topos (as I have to understand what is the recursive topos). 


Before I spend a lot of time to study them, I would like to know if there are somehow related to what I've stated above (i.e., does one of them contains in some way the computable functions between topological spaces ? is there only continuous functions between topological spaces in these Topos ? can we even talk about topological spaces in these topos ?) or are hey something completely different ?


I apologize if my question seems too general. Any answer will be appreciated :)



Effective topos and computability in topological spaces
There are many realizability toposes, of which the effective topos is just one. Each realizability topos has its own internal language in which topology and analysis can be developed to a considerable degree. Thus we get many notions of computable topology, not just one.


If you are interested in computable topology in the effective topos specifically, but would like to study the topic without first learning the machinery, you should look at Dieter Spreen's JSL paper *On Effective Topological Spaces* (doi:[10.2307/2586596](https://doi.org/10.2307/2586596),
[JSTOR](https://www.jstor.org/stable/2586596)).


If you are just getting into the topic of computable analysis, you should look at Klaus Weihrauch's [Computable Analysis: an introduction](http://books.google.com/books/about/Computable_analysis.html?id=OPolVWVFDJYC).


If you would like to understand more generally why there are many kinds of computable topology, not just one, you can read [my Ph.D. thesis](http://math.andrej.com/2000/09/20/the-realizability-approach-to-computable-analysis-and-topology/). This may also serve as a general introduction to computable analysis and topology.


