TITLE: In which spaces can one "flip" the topology?
QUESTION [13 upvotes]: This is just out of curiosity:
In which topological spaces can I relabel all closed sets as open and all open sets as closed and still obtain a valid topology?
For example, both the discrete and the indiscrete topolgy can be "flipped" for finite sets. The real topology cannot be flipped. Is there any good criterion for being a "flippable" topology?

REPLY [20 votes]: https://en.wikipedia.org/wiki/Alexandrov_topology
Your spaces are closed under arbitrary intersections. These are called Alexandrov spaces.