# Stone-Čech compactification is extremally disconnected.

If $$X$$ is a discrete topological space, one can realize its Stone-Čech compactification by means of ultrafilters.

The compactification can be characterized in terms of its universal property. I want to know if there is a "direct" proof of the fact that the Stone-Čech of a discrete space is an extremally disconnected space, by using "only" its universal property. That is, without appealing to any of its concrete realizations.

An extremally disconnected space is a space such that the closure of every open set is clopen.

If $$O$$ is open in $$\beta D$$ (where $$D$$ is discrete), let $$f: D \to \{0,1\}$$ be the function that sends points in $$O \cap D$$ to $$0$$ and all other points to $$1$$; this is a continuous function as $$D$$ is discrete so the universal property tells us that there is a (unique) $$\beta f: \beta D \to \{0,1\}$$ that extends $$f$$. As $$O$$ is open and $$D$$ is dense, $$\overline{O \cap D} = \overline{O}$$ and as $$f[O \cap D]=\{0\}$$ we get that $$\beta f [\overline{O}]= \{0\}$$ as well, and we can show that $$\overline{O}=(\beta f)^{-1}[\{0\}]$$, a clopen set.