Is there a meaningful generalization of topology?

Question

A topology T is a collection of subsets of some set where T is closed under arbitrary union and finite intersection. Do we get a meaningful generalization if instead of union and intersection, we choose other kind of constructions in the category of sets (maybe some limits, some colimits or hom-sets) such that T is closed under these constructions? For example, sigma algebras may be considered in this way. Are there any others?