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 X λ , provided that λ < χ ( p , X ) , or, respectively, that λ < cf κ . 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.