# In which spaces can one “flip” the topology?

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?

• Both of your examples are cases where all open sets are clopen, so generalizing that to all topologies whose open sets are closed is another obvious step. I have a hunch that the topology generated by left half-open intervals and the topology generated by right half-open intervals might be another example, but I didn't check the details. – rschwieb Oct 20 '17 at 13:17

Let $T$ denote the topology, then $T^c:=\{U^c:U\in T\}$ is a topology if $T$ is closed under every kind of intersection (not just finite ones).