Filters II: filter spaces

I said earlier that what convergence was the starting point of topology.  Why not take this seriously and replace topological spaces by spaces that would be defined in terms of notions of convergence directly, instead of through opens?

One bonus is that we shall obtain Cartesian-closed categories, and this will be easy!  As usual, while nets will allow us to grasp the concept, this will actually work best with filters.  See the full post.