Let $X$ be a compact Hausdorff space and $x \in X$ be a non-isolated point.
In the proof of Lemma 3.1 in "Norming $C(\Omega)$ and Related Algebras" by B. E. Johnson it seems to be asserted that the Stone–Čech compactification of $X \setminus \{x\}$ is considered as the space of all ultrafilters of non-empty relative closed sets of $X \setminus \{x\}$.
I don't know how to topologize this space and how to show that the space is the Stone–Čech compactification.
I thought that the topology is same as Stone space of Boolean algebra. However, the lack of "complements" of the algebra of non-empty relative closed sets of $X \setminus \{x\}$ makes it difficult to show the compactness.