Old answer, retrieved from the rollback dump and wikified in response to @supercooldave's lament
Don't really know much about this, but stumbled upon these a while back, they're called the Kuratowski Closure axioms and they concern a generalised 'closure' operation cl:
$A\subseteq cl(A)$
$cl(cl(A))=cl(A)$
$cl(\emptyset)= \emptyset$
$cl(A \cup B)= cl(A) \cup cl(B)$
They allow you to define a topology by declaring the closure of every set to be closed, giving continuity a pretty little characterisation of: $f$ is cts iff $f(cl(A)) \subseteq \hat{cl}(f(A))$ for all subsets $A$ (where $\hat{cl} $ is the closure operation in the target space).
The problem, of course, for taking this into a more general category would be axiom 4- straight up, old fashioned union doesn't work for, say, rings- but I'm sure with some adjoint construction in place of the union, this could describe all of your examples.
I think the reason this construction isn't exactly famous, though, is why would anyone want to put a topology on, say, the category of integral domains??? What would its homotopy type tell you??!! (Disclaimer: there may be, despite my skepticism, an incredibly interesting and useful answer to these questions...)
Edit: Just to clarify, this is on the potential 'applications' side of the question. One would hope, with the particular adjoint condition used to define a generalised 'union', that the above axioms would be satisfied by your definition and hence give a topology to a category for which such a construction is possible.