First category $ \cup_{n}\cap_{m}\cup_{k}A_{n,m,k} $

Question

Let $X$ be a topological space and let us consider nowhere dense subsets $(A_{n,m,k})_{n,m,k \ge 1}$. Is it true that $$ \bigcup_{n\ge 1}\bigcap_{m\ge 1}\bigcup_{k\ge 1}A_{n,m,k} $$ is of first category?

Answer 1

The inner union gives us sets of first category, and the intersection only makes them smaller, so all sets $\bigcap_{m\ge 1}\bigcup_{k\ge 1}A_{n,m,k}$ are also of first category, and so their union is too.

Note that this works in any $\sigma$-ideal.