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.