Is the Stone–Čech compactification of a dense $G_\delta$set $X \subset\beta N \setminus N$ homeomorphic to $\beta N \setminus N$? Here, $\beta N \setminus N$ is the complement of $N$ in the Stone–Čech compactification of $N$.
For every dense $G_\delta$ subset $A \subset$ $\beta N \setminus N$, is it true that there is a dense subset $B\subset A $ such that $\beta B=\beta N \setminus N$?

$\begingroup$ For this problem, do you mean that $\beta N\setminus N$ $\textit{is}$ the StoneCech compactification of a dense $G_{\delta}$set $X$ (i.e. $X$ is $C^{*}$embedded in $\beta N\setminus N$) or simply that $\beta X$ and $\beta N\setminus N$ are homeomorphic as you have said? $\endgroup$ – Joseph Van Name Jan 31 '14 at 17:04

$\begingroup$ @JosephVanName: that´s a good point. I should point out however that in the Corollary 1.5.4 mentioned in my answer it is actually proved that $\omega^*\setminus\{x\}$ can not be $C^*$embedded in $\beta N$ and therefore cannot be homeomorphic to $\omega^*$. $\endgroup$ – Ramiro de la Vega Jan 31 '14 at 18:32
If you consider only complements of points, which are particular dense $G_\delta$sets, the answer is independent of $ZFC$. The following results can be found in van Mill´s article in the Handbook of SetTheoretic Topology:
Corollary 1.5.4: $(CH)$ If $x \in \omega^*$ then $\omega^* \setminus \{x\}$ is not $C^*$embedded in $\omega^*$.
and
Corollary 2.5.3: It is consistent that for some $x \in \omega^*$ we have that $\beta (\omega^* \setminus \{x\})=\omega^*$.
I don´t know if it is possible to prove in $ZFC$ that there is a dense $G_\delta$subset of $\omega^*$ which is not $C^*$embedded in $\omega^*$.