TITLE: What are interesting families of subsets of a given set?
QUESTION [35 upvotes]: 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?

REPLY [7 votes]: Another ultrafilter cousin is the concept of a majority
space. This is a family $M$ of nonempty subsets of $X$,
called the majorities, such that any superset of a
majority is a majority, every subset of $X$ or its
complement is a majority, and if disjoint sets are
majorities, then they are complements. A strict majority
space has $Y\in M\to Y^c\notin M$, and otherwise they are
called weak majorities. A vast majority space is closed
under finite differences in majorities. There are other
various overwhelming majority concepts.
The main point is that the majority space concept
generalizes the ultrafilter concept by omitting the
intersection rule. Every ultrafilter on $X$ is a majority
space. But there are others. For example, on a finite set,
one may take the the subsets with at least half the size, and of course this situation motivates the voting theory terminology.
On an infinite set $X$, one can divide it into a finite odd
number of disjoint pieces $X_i$, each carrying an
ultrafilter $\mu_i$ on $X_i$, and then saying that
$Y\subset X$ is a majority if for most $i$ one has
$Y\cap X_i\in\mu_i$. This produces a vast majority on
$X$ that is not an ultrafilter.
Eric Pacuit has investigated majority logic, and I recall that Andreas Blass has some very interesting work showing that it is consistent with ZFC that every majority space derives from ultrafilters in a simple way.