If we have a subset $Y$ of a complete metric space $X$ such that $Y$ is a $G_{\delta}$, then how would I go to show that $Y$ satisfies the Baire category theorem?
I am trying to get there by showing that any meagre subset $Z \subseteq Y$ has $int(Z)=\emptyset$ but do not know how to use the fact that $X$ is complete in my proof... Any help or ideas? Greatly appreciated.