Let $X$ be a completely metrizable space and let $A\subset X$ be a co-meager subset.
Since $X$ is Baire, we know that there is some dense $G_\delta$ set contained in $A$, say $G\subset A$. But since $X$ is complete and $G$ is $G_\delta$, then $G$ is complete. Hence, as a complete subspace of another complete subspace, $G$ is closed.
But since $G$ is dense, $G=X$ which implies that $A=X$. So we conclude that any co-meager subset of a completely metrizable space, has to be the space itself. This is not true but I can't see where I made a mistake.
Can somebody help me? Thanks