We know that Baire Category Theorem implies that in a complete metric space, the countable intersection of open and dense sets is nonempty and actually dense itself.
But it is clear that a countable intersection of open sets need not be open.
So, can you find an example of a non-open set that is a countable intersection of open and dense sets in a complete metric space?
Best Answer
Sure:
For each rational number $ r$, let $U_r=\Bbb R\setminus\{\,r\,\}$. Then the set of irrational numbers is the intersection of the sets $U_r$.