\(\pi\)-character and depth in scattered Boolean spaces (Q1612228)

J. D. Monk raised the question of the relationship between the \(\pi\)-character and depth in compact scattered spaces. The question turned out to be rather interesting in the reviewer's opinion. A space is scattered if every subspace has an (relatively dense set of) isolated point(s). In such a space the \(\pi\)-character (\(\pi\chi(x)\)) of a point turns out to be simply the least cardinal of a set of isolated points which has the point in its closure. The \(\pi\)-character of a space \(X\), \(\pi\chi(X)\), is the supremum of the \(\pi\)-characters of the points. The depth of a space is the supremum of all cardinals for which there is a strictly increasing chain of that order type consisting of clopen subsets. There is an example of a compact scattered space with \(\pi\chi=\omega_1\), while the depth is countable. The author shows that there is no such space in which \(\pi\chi=\chi\) for each point of countable \(\pi\)-character. Another known result, by contrast, is that if there is a point \(x\) in a compact scattered space \(X\) with \(\pi\chi(x)=\kappa\) a regular cardinal greater than \(\omega_1\), then the depth of \(X\) is at least \(\kappa\). The author generalizes this result to the case where \(\kappa\) is a singular cardinal of cofinality \(\omega\) and only needs to assume that \(\pi\chi(X)=\kappa\) rather than there is an \(x\) with \(\pi\chi(x)=\kappa\).