I want to complete something said by Andrew Stacey above: like him I think that the only reason to motivate the use of the open sets it's because they are more easy to use. Topology is the study of property preserved by invertible continuous transformation (following the Erlangen program): this definition clearly need the notion of continuity, I've always thought of continuity as the relation of proximity of points, so the first thing to do topology is to define the notion of proximity and neighbourhood are most natural way to do so (at least for me). Anyway deal with neighbourhood is more complex than working with open set, for example the definition of topology with neighbourhood require five axioms while classical definition with open sets require just three axioms, so while it seems more natural the study of topology via neighbourhood it is more convenient dealing with open sets which allow to simplify proofing work.

I hope this answer my help.

I want to complete something said by Andrew Stacey above: like him I think that the only reason to motivate the use of the open sets it's because they are more easy to use. Topology is the study of property preserved by invertible continuous transformation (following the Erlangen program): this definition clearly need the notion of continuity, I've always thought of continuity as the relation of proximity of points, so the first thing to do topology is to define the notion of proximity and neighbourhood are most natural way to do so (at least for me). Anyway dealing with neighbourhoods is more complex than working with open set, for example the definition of topology with neighbourhoods require five axioms while classical definition with open sets require just three axioms. So while it seems more natural the study of topology via neighbourhoods it is more convenient dealing with open sets which allow to simplify the proofs.

I hope this answer my help.