Power homogeneous compacta and the order theory of local bases (Q624414)

A space \(X\) is power homogeneous iff some \(X^{\kappa}\) is homogeneous. In recent years there have been a number of papers regarding cardinal invariants of power homogeneous spaces. This paper considers the cardinal invariant split\(_{\kappa}(p, X)\), defined below, which generalizes the notion of Noetherian type, not defined in this review. Given \(p \in X\), split\(_{\kappa}(p, X)\) is the least cardinal \(\lambda\) so that there is a family \(\{u_{\alpha}: \alpha < \kappa\}\) of open neighborhoods of \(p\) such that \(p\) is not in the interior of any \(\bigcap_{\alpha \in E}u_{\alpha}\) where \(|E| = \lambda\). In particular, if the character of a point \(p\) is \(\kappa\) and \(p\) does not have a finite local base, then the local Noetherian type of \(p\) is the \(\kappa\)-splitting number of \(p\). A point is flat iff its local Noetherian type is \(\omega\), which translates to: \(p\) is flat iff split\(_{\chi(p, X)}(p, X) = \omega\). A space \(X\) is flat iff all of its points are. All compact power homogeneous spaces known to the authors are flat. What about compact power homogeneous spaces we don't know about? The main theorem in this paper is that, under GCH, if \(X\) is a power homogeneous compact space, and \(\sup_{p \in X}\chi(p, X) = \operatorname{cf} \chi(X) > d(X)\) then \(X\) contains an open set of flat points. Along the way a number of theorems are proved about Noetherian types and split\((p,X)\) in partial box products, i.e., spaces of the form \(\prod^{(\kappa)}_{i \in I}X_i\) whose basic neighborhoods are sets of the form \(\prod_{i \in X}u_i\) where fewer than \(\kappa\) many \(u_i \neq X_i\).