What is a geometric structure? Every elementary book on abstract algebra usually begins with giving a definition of algebraic structures; generally speaking one or several functions on cartesian product of a point-set to the set. My question is this: Is there a property that unifies different geometric structures like topology(I consider it a geometric structure), differential structure, incidence structure and so on? Can one say a geometric structure on a set one way or another involves a subset of its powerset ?

It is known that, for a (compact) topological space, the continuous functions into $\mathbb{C}$     characterize the topology on the space. As far as I know similar statements hold for smooth manifolds (using smooth functions) and algebraic varieties (using polynomials). So one possible answer is that a geometric structure is an algebra of functions on your space.
