How does one prove the following result (Engelking, exercise 3.9.A):
If a Cech-complete space $X$ is a subspace of a Hausdorff space $Y$, then there exists a $G_\delta$ set $Z \subseteq Y$ such that $X=\overline{X}\cap Z$.
In other words, if $X$ is a Cech-complete subspace of a Hausdorff space, it is a $G_\delta$ in its closure.
Related and Possibly Relevant Facts:
For the record, a Cech-complete space is a Tychonoff space that is a $G_\delta$ in its Stone-Cech compactification (or equivalently, in any of its compactifications).
Cech-complete spaces generalize locally compact spaces in the sense that locally compact spaces are open in any of their compactifications.
The Baire Category Theorem holds in Cech-complete spaces.
The result to be shown parallels the fact that a locally compact subspace of a Hausdorff space is open in its closure.