Local compactness does not always imply spatiality
The Priestley spaces of Examples 3.5, 4.6 and 4.10 in Local compactness does not always imply spatiality with Guram Bezhanishvili, Ranjitha Raviprakash, and Anna Laura Suarez (arXiv). In Examples 3.5 and 4.10, XB is the Stone space of a complete atomless Boolean algebra B: it is extremally disconnected and has no isolated points.
Example 3.5. M = B × B, where B is a complete atomless Boolean algebra and □(a, b) = (a ∧ b, b). The Priestley dual of O(M) is XB × {0 < 1}, a collection of two-element chains: XB stacked on top of itself, each point of the lower copy below its copy in the upper one.Example 4.6.Work in progress L = Ω(S) × 𝒫(D) with S = ℚ ∪ {∞}, so XL = XΩ(S) ⊔ βD. This is not the whole space: beside βD it shows one closed piece of XΩ(S), around a rational q at which ℚ is not locally compact. The row remainders An* lie above p∞, the diagonal part Aωd above pq, and Aω∞ above both.Example 4.10. A complete atomless Boolean algebra B with a new top adjoined. Its Priestley space is XB ⊔ {p}, where p is an isolated least point and the points of XB are mutually incomparable.