Topological Modules are defined here Wikipedia. My question is can we define notions like injective modules or projective modules? Can we define $Tor$ and $Ext$ functors? I have tried hard to find references but in vain.
Edit: I would also like to know if there is any universal coefficient theorem for topological modules.
In general, the category of topological modules for a topological ring does not have enough projectives, relative to all the short exact sequences you might want - i.e. a submodule with the subspace topology, and a quotient module with the quotient topology. I don't know about injectives but I suspect the answer is the same.
But by restricting the class of short exact sequences, we can give the category an 'exact structure' for which there are enough projectives. In my paper, https://arxiv.org/abs/1703.00569, I consider an exact structure for which the projectives are the free modules on disjoint unions of compact Hausdorff spaces. You can define total derived functors here. But beware of taking the homology of the total derived functors: things don't work as well in an exact category as in an abelian one. Things get a little involved, so follow the link for the details.