The usual starting point of both Topology and Measure Theory is the definition of a family of subsets of a set $S$.
Indeed, one defines a topology on $S$ to be a family of subsets including the empty set $\emptyset$ and $S$ itself and which is closed under arbitrary unions and finite intersections. These are the open sets. (One can of course also define a topology by stipulating which are the closed sets, which are now closed under finite unions and arbitrary intersections.)
In Measure Theory one starts by defining on $S$ the notion of a $\sigma$-algebra, which is a collection of subsets again including $S$ and which is closed under complementation and countable unions, so in particular it also includes $\emptyset$ and is closed under countable intersections. The subsets in the $\sigma$-algebra are the measurable sets.
When I learnt these subjects I was always intrigued by the similarity of both definitions. This suggests other family of subsets of a set $S$ defined by demanding that both $\emptyset$ and $S$ belong to the family and that the family be closed under some operations.
Are there any interesting families of subsets, other than topologies and $\sigma$-algebras, which can be defined in this way? And if so, to which areas of mathematics are they germane?