Probably $\beta \mathbb N$ is not an absolute retract (is there an easy argument for this?), but I'd be interested to know what happens in the class of extremally disconnected (compact) spaces. Is it an absolute retract therein?



Is the Čech–Stone compactification of the integers always a retract of an extremally disconnected space?
$\beta\mathbb{N}$ is not a retract of a Tychonoff cube because it is not connected; it also not a retract of a Cantor cube, not even a continuous image, see problem 3.12.12 in Engelking's book.


It is an absolute retract for ED spaces: if it is embedded in the compact ED space $X$ then $\mathbb{N}$ is relatively discrete subspace of $X$ and we have pairwise disjoint open sets $O\_n$ in $X$ with $O\_n\cap\beta\mathbb{N}=\{n\}$.
Then the closure $K$ of the union $O=\bigcup\_nO\_n$ is clopen and by extremal disconnectedness $K=\beta O$. Now extend the retraction $r$ from $O$ onto $\mathbb{N}$, defined by $r(x)=n$ if $x\in O\_N$, to $\beta r:K\to\beta\mathbb{N}$. As $r$ is the identity on $\mathbb{N}$ the extension $\beta r$ is the identity on $\beta\mathbb{N}$.


