There are classical undecidable problems in topology, such as deciding whether two manifolds are homeomorphic. These problems require the manifold to be presented as a simplicial complex (a finite object). However, not all manifolds are triangulable. I will describe an approach where the manifold is instead presented as the completion of a countable metric space. So far, these methods have only been applied to surfaces. The main result here is that given a metric space which is homeomorphic to a surface, we can construct an atlas and triangulation, and hence obtain the standard classification of surfaces, in an arithmetic way. (I will explain in my talk what arithmetic means here, but one should take it to mean "not too complicated".)