Why is the open set definition of topology used I am studying topology, and trying to get some intuition for why the typical open set formulation (closed under arbitrary union, finite intersection) was selected as the definition.
 A: Unfortunately I don't know the precise origin of the definition of a topology
as a collection of "open" sets satisfying the standard axioms.  
Nevertheless, one should note that it is pretty easy to move between the open set definition and the closed set definition (just taking complements), so it's probably not worth putting too much weight on which of the two structures is
singled out in the standard definition.
You might be interested to know that 
in some contexts, the topological structure is naturally defined in terms of closed sets rather than open sets.  For example, the Zariski topology on
algebraic sets over a field, or on the Spec of a (commutative, with unity) ring,
is naturally defined in terms of the closed sets.  The closed sets are defined
directly in terms of ideals, while a general open set doesn't really have a direct description except as the complement of some closed set; only the distinguished opens admit a direct description (and while these form a base for the topology, but not every open set need be distinguished).
