A question raised via email by Dave Long (one of my partners-in-crime on Cleese) has prompted these thoughts.
There is an inherent circularity to think of topologies as collections of open sets because it is the topology that defines what an open set is to start with. There's nothing inherent in an open set that makes it "open" apart from the fact it is a member of the topology.
In sets with more structure that enable you to define openness in terms of that additional structure, openness still comes down to the choice of topology that the additional structure is implying.
For example, if you choose a distance function for a metric space, you've implicitly chosen the topology. So while the open sets can be explicitly defined by the distance function in that case, the very choice of the function assumes a particular underlying topology.
UPDATE: next post
The original post was in the category: poincare_project but I'm still in the process of migrating categories over.