1. 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 $N$.

  2. 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 Stone-Cech 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 Set-Theoretic Topology:

Corollary 1.5.4: $(CH)$ If $x \in \omega^*$ then $\omega^* \setminus \{x\}$ is not $C^*$-embedded in $\omega^*$.


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 $\omega^*$.


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.