In which spaces can one "flip" the topology?

Question

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

score 20 · Accepted Answer · edited Oct 20 '17 at 17:26

20

https://en.wikipedia.org/wiki/Alexandrov_topology

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

edited Oct 20 '17 at 17:26

user2357112

2,401

answered Oct 20 '17 at 13:18

1

Ooo, I hadn't heard this particular term before... :) – rschwieb Oct 20 '17 at 13:20
4

Also, any finite space as a special case. They have an unusually strong homotopy theory, much of it worked out in the 60s. – Randall Oct 20 '17 at 13:40
@Randall every finite space is Alexandrov. – Oct 20 '17 at 13:41
1

Yeah, I know that. I'm pointing out a particularly successful special case. – Randall Oct 20 '17 at 13:42

score 6 · Answer 2 · answered Oct 20 '17 at 13:15

6

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).

answered Oct 20 '17 at 13:15

Yannik

628

In which spaces can one "flip" the topology?

2 Answers2