Metacompact subspaces of products of ordinals (Q2750917)

The author provides a characterization of (\(\lambda\)-)metacompactness for subspaces of finite products of ordinals. This characterization is in the spirit of the Engelking-Lutzer characterization of metacompactness in ordered spaces [\textit{R. Engelking} and \textit{D. Lutzer}, Fundam. Math. 94, 49-58 (1977; Zbl 0351.54014)]. The key is to define an appropriate notion of stationary set in products of ordinals. Given regular cardinals \(\kappa_1\leq\kappa_2\leq\cdots\leq\kappa_n\) one defines \(A\subseteq\kappa_1\times\kappa_2\times\cdots\times\kappa_n\) to be \((\kappa_1,\kappa_2,\ldots,\kappa_n)\)-stationary if in the tree \(T(A)=\{a\mathbin{\upharpoonright}i:a\in A, i\leq n\}\) every node \(s\) below the top level has \(\{\xi:\langle s,\xi\rangle\in T(A)\}\) stationary in \(\kappa_{i+1}\), where \(i=|s|\). The main result of the paper states: if \(X\subseteq\alpha^n\) for some ordinal~\(\alpha\) then \(X\)~is \(\lambda\)-metacompact iff \(X\)~has no closed set homeomorphic to a \((\kappa_1,\kappa_2,\ldots,\kappa_n)\)-stationary set with \(\kappa_1<\lambda\). The case \(\lambda=\aleph_1\) says that all finite products of ordinals are hereditarily countably metacompact and the case \(\lambda=\infty\) shows that the Engelking-Lutzer characterization is valid for spaces of the type under consideration.