Proposition 5

In a Baire space $X$, the complement of a meagre set is a Baire space.

Let $A$ be a meagre set in $X$; then its complement $Y = \complement_A$ in $X$ is dense in $X$. Let $B$ be a meagre set relative to $Y$; $B$ is also meagre relative to $X$, hence $A \cup B$ is meagre relative to $X$. Hence the complement of $A \cup B$ in $X$, which is also the complement of $B$ in $Y$, is dense in $X$ and a fortiori dense in $Y$. Hence $Y$ is a Baire space.