Completely metrizable space in Baire space I know that any subset $A$ of a complete metric space $(X,d)$ is completely metrizable iff it is $G_\delta$ subset of $(X,d)$. I want to know whether this result hold in case of any Baire space or not, i.e. can we say that any $G_\delta$ subset of a Baire space is completely metrizable? If yes how ? If no please provide a counter example . 
 A: Completely metrisable certainly not: if we start with a non-metrisable Baire space (of which there are many, e.g. many locally compact Hausdorff spaces) there is no reason to expect that a $G_\delta$ subset of it would be metrisable, let alone completely metrisable: in particular $X$ is a $G_\delta$ subset of itself! So all
spaces in this list from $\pi$-base are examples that way. 
So if completely metrisable is out, can we preserve Baire-ness? Well, a dense $G_\delta$ subset of a Baire space is again Baire but this fails for non-dense sets, as shown by the (closed, so $G_\delta$, as we're in a metric setting) non-Baire $A=\mathbb{Q}\times \{0\}$ inside the Baire space $(\mathbb{R} \times [0,+\infty)) \setminus (\mathbb{P} \times \{0\})$ (subspace of $\mathbb{R}^2$) where $\mathbb{P}$ denotes the irrationals.
A Baire subset of a Baire space need not be a $G_\delta$ and it can be even non-Borel, as the Bernstein subsets of $\mathbb{R}$ discussed here e.g. 
So Baireness is nice but doesn't behave as nicely as the much stronger complete metrisability; in-between we have the notion of Čech-completeness, which means that $X$ is a Tychonoff space that a $G_\delta$ in one of its Hausdorff compactifications. A Čech-complete space is Baire and the property is preserved to closed and $G_\delta$ subsets. Moreover, a metrisable space is completely metrisable iff it is moreover Čech-complete. So this notion is the one in general topology that probably most approximates completely metrisable ones. It also includes all locally compact Hausdorff spaces (a common class of Baire spaces in practice).
