It is well known that under CH every totally-disconnected compact F-space of weight at most $\omega\_1$ can be embedded into the remainder $\omega^\*=\beta\omega\setminus\omega$ of the Cech-Stone compactification of $\omega$. My first two questions are as follows:


**Question 1:** Under which other axioms (or weaker: in which models of set theory) does this fact also hold? What are known consistent counter-examples to this fact?


**Question 2:** Can this fact be consistently generalised to higher weights, e.g. under GCH?


My last question concerns embedding of some special separable spaces into $\beta\omega$.


**Question 3:** Let $\kappa$ be an infinite cardinal number and $A$ a closed separable subspace of $\beta\kappa$, where $\kappa$ is endowed with the discrete topology. Can $A$ always be embedded into the space $\beta\omega$? And what in the case when $A$ is countable but not necessarily closed in $\beta\kappa$?


Thank you in advance for the answers or even useful hints!



Embeddability into $\beta\omega$ and $\omega^*$
Answer to 1: In [On closed subspaces of $\omega^\*$](https://doi.org/10.1090/S0002-9939-1993-1152978-5) (Proc. AMS, 1993) it is shown by Dow, Frankiewicz and Zbierski that in the $\aleph\_2$-Cohen model every compact zero-dimensional $F$-space of weight at most $\mathfrak{c}$ is embeddable on $\omega^\*$.


Answer to 3: yes. If $A$ and $B$ are countable subsets of $\beta\kappa$ that are separated ($\overline{A}\cap B=\emptyset=A\cap\overline{B}$) then they are contained in disjoint clopen sets; this shows that separable subsets of $\beta\kappa$ are extremally disconnected and hence embeddable in $\beta\omega$. To prove the first claim (which is most likely well known) enumerate $A$ and $B$ as $\{a\_n:n\in\omega\}$ and $\{b\_n:n\in\omega\}$ respectively. Choose subsets $U\_n$ and $V\_n$ of $\kappa$ such that $U\_n\in a\_n$ and $V\_n\in b\_n$ for all $n$ as well as $U\_n\notin b\_m$ and $V\_n\notin a\_m$ for all $m$ and $n$. Next put $X\_n=U\_n\setminus\bigcup\_{i<n}V\_i$ and $Y\_n=V\_n\setminus\bigcup\_{i\le n}U\_i$. Finally let $X=\bigcup\_nX\_n$ and $Y=\bigcup\_n Y\_n$; then $X\cap Y=\emptyset$, and $X\in a\_n$ and $Y\in b\_n$ for all $n$.


