@godelian This seems promising, and your example interests me. I can (sort of) form a mental picture of an $η_1$ totally ordered set, like the set of functions $ω_1 \to \{-1,0,+1\}$ that are eventually 0, ordered lexicographically, and I think $η_1$ means the same as $\aleph_1$-saturated for the order structure. Can you describe what the topology you are referring to is, more concretely, in this example (e.g., how it relates to the order topology), which would make it extremally disconnected? (Please don't hesitate to make an answer out of this.)
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