Suppose I have an allergy to non-Hausdorff spaces but I really want to do, say, arithmetic geometry. I wonder if there is some perverse way I could develop scheme theory that would accomodate for my needs. 


I am not sure what is the best way to formalize this question but here is an attempt. Does there exist a functor from schemes to the category of Hausdorff topological spaces/continuous maps that is one of the following


* full and faithful
* faithful and admits a left adjoint
* full and admits a left adjoint
* faithful and admits a right adjoint
* full and admits a right adjoint?


P.S. I believe that there is a faithful functor from schemes to sets (take the union of the underlying set and all the stalks and do some power-set acrobatics) and there is a faithful functor from sets to Hausdorff spaces (discrete topology) so at least a faithful functor should exist. 



Doing scheme theory with Hausdorff spaces
It's not clear to me that any of these properties are even attained by a functor from schemes to *all* topological spaces. For instance, the functor sending a scheme to its underlying space isn't full or faithful, nor does it preserve limits (in particular, it doesn't preserve the terminal object or binary products) -- maybe it preserves colimits? Anyway, that makes me doubt that these questions are the right direction to look for a "Hausdorff theory of schemes".


Maybe a more fruitful place to look for a "Hausdorff theory of schemes" is to consider the boolean algebra generated by the Zariski-open sets (as a subalgebra of the boolean algebra of closed points of a scheme $X$). This is the boolean algebra of constructible sets, which is a totally disconnected Hausdorff topology. It's often considered in model-theoretic approaches to algebraic geometry, as far as I understand.


