Proposition 4

If every point of a topological space $X$ has a neighbourhood which is a Baire space, then $X$ is a Baire space.

Let $A$ be a non-empty open set in $X$, let $x \in A$ and let $V$ be an open neighbourhood of $x$ which is a Baire space. If $A$ were meagre in $X$, then $V \cap A$ would be meagre in $V$ and open in $V$, which is contrary to hypothesis.