If $Y$ is a complete metrizable dense subset of a topological Hausdorff space $X$...then is $X$ a normal space?
I tried to solve it using the fact that every point $x$ in a closed in a Hausdorff space is a limit of a sequence of points in the space. In particular, $Y$ is dense, so there must be a sequence of points of $Y$ converging to $x$. And since $Y$ is completely metrizable, we should have that the point belongs to $Y$. But this seems wrong to me since Y=X, if this is true. Even so, it seems to me that such a fact is true...could someone give me a proof of such a fact or a counterexample?