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".
The #4 most popular answer is "1 + the product of the first n prime numbers is always prime".
The #2 most popular answer is "the trace of a product of any number of matrices is independent of their order"
Here are verifications that these are indeed false beliefs. :)