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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?

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  • $\begingroup$ 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. $\endgroup$ – rschwieb Oct 20 '17 at 13:17
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https://en.wikipedia.org/wiki/Alexandrov_topology

Your spaces are closed under arbitrary intersections. These are called Alexandrov spaces.

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    $\begingroup$ Ooo, I hadn't heard this particular term before... :) $\endgroup$ – rschwieb Oct 20 '17 at 13:20
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    $\begingroup$ Also, any finite space as a special case. They have an unusually strong homotopy theory, much of it worked out in the 60s. $\endgroup$ – Randall Oct 20 '17 at 13:40
  • $\begingroup$ @Randall every finite space is Alexandrov. $\endgroup$ – user223391 Oct 20 '17 at 13:41
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    $\begingroup$ Yeah, I know that. I'm pointing out a particularly successful special case. $\endgroup$ – Randall Oct 20 '17 at 13:42
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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).

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