Are there any sets where it is an open question whether the set is open (or closed)?

Question

Proving whether a set is open or closed in a topological space can be difficult at times, so I'm curious if there are any sets where it is an open question whether they are open or closed.

Obviously, you can construct trivial questions like $S=\mathbb{R}$ if some open question is true, but of course that is not what I am looking for.

Answer 1

Nobody knows if the set of all reciprocals of Mersenne primes is closed. Or take the set of all reciprocals of odd perfect numbers; it is conjectured that it's open, there is a weaker conjecture that it's closed, but nobody knows.

Answer 2

It's easy to make a contrived one. Take any open yes-or-no question, let your set be $(0, 1)$ if the answer is no and $[0,1]$ if the answer is yes (as subsets of $\mathbb R$).

Answer 3

Let $\gamma$ be the Euler-Mascheroni constant. The set $S=\{x \in \mathbb{Q}|x \le \gamma\}$ is an open subset of $\mathbb{Q}$ if and only if $\gamma$ is irrational, which is an unsolved problem. So the openness of $S$ is open.