**Motivation**

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.

**Question**

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?

algebraof sets, rather than a $\sigma$-algebra, is also natural and important. This is already touched upon in Gjergji Zaimi's answer below. – Pete L. Clark Dec 30 '09 at 11:18set with structure, instead you should study categories. And, although I would deplore that response, I admit that there is a point there. Much of the point of defining sigma-algebra is in order to define "measurable function" ... and much of the point of defining topology is in order to define "continuous function". – Gerald Edgar Jan 31 '10 at 18:28