This is just a curiosity and might be a dumb question, but is there any area of mathematics where the notion of infinite series (let's say with regard to a not necessarily abelian group or even monoid structure) makes sense (and is useful) without having a topology and hence a notion of convergence, and even without being able to topologise the space in question so that all the series you want converging converge to what you want? I am aware that one of the main purposes of topology is precisely to give meaning to the idea of convergence, so the answer seems negative, but I might be wrong.
Also, another possibly related question I have is: given a collection of sequences (not necessarily series now) that converge by definition in some set, is it possible to define a topology on the set which preserves these convergences (and preferrably only these, maybe adding more limit points to some sequences?). I'm thinking for example of simple convergence of functions as a palpable example. Also if the answer is negative, are there any conditions on the set of converging sequences so that this works etc.?