Power homogeneous compacta and the order theory of local bases

We show that if a power homogeneous compactum X has character κ+ and density at most κ, then there is a nonempty open U⊆X such that every p in U is flat, “flat” meaning that p has a family F of χ(p,X)-many neighborhoods such that p is not in the interior of the intersection of any infinite subfamily of F. The binary notion of a point being flat or not flat is refined by a cardinal function, the local Noetherian type, which is in turn refined by the κ-wide splitting numbers, a new family of cardinal functions we introduce. We show that the flatness of p and the κ-wide splitting numbers of p are invariant with respect to passing from p in X to 〈p〉α<λ in Xλ, provided that λ<χ(p,X), or, respectively, that . The above <χ(p,X)-power-invariance is not generally true for the local Noetherian type of p, as shown by a counterexample where χ(p,X) is singular.