The #1 most popular question on MO is "Examples of common false beliefs in mathematics".
The #8 most popular answer is "if a set is open, then it is not closed".
https://mathoverflow.net/a/23580/4600

[Here][1] is a verification that this is indeed a false belief. :)


  [1]: https://live.lean-lang.org/#code=import%20Mathlib.Topology.Clopen%0A%0Aexample%3A%20IsOpen%20(%E2%88%85%20%3A%20Set%20Bool)%20%3A%3D%20trivial%0Aexample%3A%20IsClosed%20(%E2%88%85%20%3A%20Set%20Bool)%20%3A%3D%20isClosed_empty%0A