Baire’s Category Theorem Example I wish to find an example showing that Baire’s Category Theorem may not work when applied to uncountable collections of open and dense subsets. My first thought was the rationals bu the rationals is not considered an open set right?

Consider $X=\Bbb R$ with the usual topology. It's a complete metric space.
Define $U_a=\Bbb R-\{a\}$ for each $a\in \Bbb R$. Dense and open. But
$\bigcap_{a\in\Bbb R}U_a=\emptyset$ is certainly not dense.
