What are the most interesting topological spaces that satisfy the weakest separation axioms?
When I say interesting I mean in the sense that is beyond merely underestaning topological spaces better. I'm looking more for applications in different fields that use such spaces.
Clearly metrizable spaces have huge relevance. For normal but not metric spaces I can only think of order topologies of large ordinals.
According to Munkres, it seems that it will be hard to find good examples for non-Hausdorff spaces, though I think he is impliying that there are good examples for spaces that do satisfy Hausdorff but not more.
So - I'm curious to hear of different proofs and applications in various fields that use topological space which have a weak separation - the weaker the better.