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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.

PatrickR
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  • The equivalence is only for Hausdorff compactifications (e.g. the Alexandroff extension isn't Hausdorff if the space isn't locally compact, noncompact, and Hausdorff). The Stone-Cech compactification satisfies a universal property among Hausdorff compactifications, thus the equivalence of the definitions. Then probably the result is somehow related to the strong relationship that compact Hausdorff spaces have with closed subspaces of completely regular spaces. https://topospaces.subwiki.org/wiki/Compact_Hausdorff_space – Chill2Macht Feb 14 '22 at 03:13
  • At least, what made me suspect as much is the following quote from https://encyclopediaofmath.org/wiki/Complete_space : "For topological spaces, the requirement of absolute closure (i.e. closure in any space containing it) leads to compact spaces if one restricts oneself to the class of completely-regular Hausdorff spaces: Those spaces and only those spaces have this property." I am probably wrong, but given all of those facts it really sounds like $\bar{X}$ is supposed to be compact Hausdorff -- at the very least, the result should follow if you can show $\bar{X}$ is compact Hausdorff. – Chill2Macht Feb 14 '22 at 03:15
  • Probably also relevant according to this answer to a related question: https://math.stackexchange.com/a/3832283/327486 namely (section? theorem?) 3.9.2 of Engelking's book General Topology (1989 2nd ed.) "[That closed subspaces of Cech-complete spaces are Cech-complete] is true, but needs another characterisation of Čech-complete spaces in terms of closed subsets with the FIP that Engelking shows in 3.9.2." FIP here seems to mean "finite intersection property" https://math.stackexchange.com/questions/1517076/compactness-of-the-stone-%C4%8Cech-compactification-by-ultrafilters – Chill2Macht Feb 14 '22 at 03:25

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