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A partial answer to the question about the Gleason cover of the unit interval: it can be found everywhere in every compact extremally disconnected space. The point is: in a compact extremally disconnected space the closure of every countably infinite relatively discrete subset is (homeomorphic to) $\beta\mathbb{N}$ and $\beta\mathbb{N}$ contains homeomorphic copies of all compact extremally disconnected spaces of weight $\mathfrak{c}$ (or less), see this paper for example.