What are the 'spaces' whose structure is defined by a collection of subsets I am trying to recount which 'space' structures are defined by a collection of subsets. Given a set $X$ the only two structures I know are:
(1) $\mathcal{T}\subseteq \mathcal{P}(X)$ is a topology on $X$, and $(X,\mathcal{T})$ is a topological space.
(2) $\mathcal{F}\subseteq \mathcal{P}(X)$ is a $\sigma$-algebra on $X$, and $(X,\mathcal{F})$ is a measurable space.
Are there any more such famous structures given by a collection of subsets? 
 A: In measure theory there are often technical such structures (semi-algebra/algebra on a set $X$, $\pi$-system, $\lambda$-system) that are used in the development of the theory, but are rarely studied independently, so I'll ignore them...
A bornological space ("bounded sets") is another one occurring in analysis.
A more geometric one is abstract convexity structures (where we have an axiomatisation of convex sets, but there are also metric convexities, order convexities etc. see the standard reference book by van de Vel for much more info. These are closed under arbitary intersections, contain $\emptyset$ and $X$, and are closed under all up-directed unions, to summarise the axioms. It's very much in the spirit of topology (separation axioms etc.), IMHO it's a nice theory that deserves more attention.
Stretching it a bit, (undirected) graphs are sets with special "doubleton sets" (the edges), but these have no special axioms. One can generalise to multigraphs and incidence structures and matroids (the "independent" subsets). The last one are a proper example in your sense, I think.
Maybe others can come up with more than bornological spaces, convexity structures and matroids.
A: The most important "space-like" structure defined on subsets in the realm of combinatorial commutative algebra is the (abstract) simplicial complex. These correspond in a natural way to square-free monomial ideals. This correspondence unleashes a bevy of tools from commutative algebra to study the combinatorics of simplicial complexes and vice versa. 
As hinted at by @Henno, matroids form a particular family of simplicial complexes (via their independent sets). Greedoids are a generalization of matroids that are "spaces" defined on power sets of a given (finite) set that behave well with respect to the greedy algorithm. One class of greedoids are the antimatroids which are complementary combinatorial objects to convex geometries. 
What all of these examples have in common is that they are combinatorial abstractions of some classically studied structure usually embedded in some (typically Euclidean) geometry. Simplicial complexes are abstractions of triangulated topological spaces; matroids capture all of the (linear) dependencies between vectors in a given finite subset of a vector space; convex geometries generalize shellings of polytopes.
