# Limits of Baire Spaces

A Baire space is a topological space in which intersection of open dense sets is dense.

Two of the ways in which we could obtain a Baire space is to have a complete metric space or a locally compact Hausdorff space.

Completeness does not seem to pass through limits, for example, $\{x\in L^{\infty}: ||x||\geq 1/n\}$ does not remain complete in the limit $n\to\infty$.

Question: Is it true that (countable) union of Baire spaces is Baire?

I cannot appeal to completeness because it might not hold under countable unions.

This is not true, since $$\mathbb{Q} =\bigcup_{q\in\mathbb{Q}} \{q\}$$ and every $\{q\}$ is a Baire space.