How to proof that these statements are equivalent:
- Every intersection of countably many dense open sets is dense.
- The interior of every union of countably many closed nowhere dense sets is empty.
Moreover, why does it follow for Baire Spaces and why is it strictly weaker that:
- $X=\bigcup_{k=1}^\infty A_k\quad\Rightarrow\quad\exists k_0\in\mathbb{N},x_0\in X: \operatorname{cl}(A_{k_0})\in\mathcal{N}_{x_0}$
...in words, the space is not meagre iff whenever it is a given by a countable union of subsets then at least one of them is large.