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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.

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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.

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