[Math] What are interesting families of subsets of a given set

Question

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.

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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?

Answer 1

Even families of subsets closed under unions are interesting. The following conjecture of Peter Frankl has been open for 31 years:

Let $A$ be a finite set, and let $\mathcal{F}$ be a collection of subsets of $A$, not all empty, such that the union of any two sets in $\mathcal{F}$ is also an element of $\mathcal{F}$. Then there exists an element belonging to at least half the sets in $\mathcal{F}$.