In general one can define a topological space either in terms of its open sets or in terms of its closed sets, however it seems that depending on the context the "closed sets" might make more intuitive sense, for example suppose that some class of geometric structures is closed under isomorphisms with respect to arbitrary intersections and finite unions. Now if we look at these as the open sets of a topology on some specific sets by taking complements it may be that this isomorphism between the structures no longer works as the complements are relative with respect to the points, thus it would make more sense to study these as closed sets rather then complementing them and looking at them as open sets.

Of course though it is simpler to adopt one of the two representations when defining a bunch of associated terminology otherwise you'll end up having to do this twice, with roughly the same definition only you're running around complementing every set in the corresponding definitions. Is that what this is just an arbitrary convention to save space i.e. people just use a single notion to save time?