On seminormal functors and compact spaces of uncountable character (Q387198)

The definition of a \textit{seminormal functor} acting in the category \textit{Comp} of all compact spaces and their mappings may be found in [\textit{V. Fedorchuk} and \textit{S. Todorčević}, Topology Appl. 76, No. 2, 125--150 (1997; Zbl 0913.54005)]. In what follows, given \(X\) a topological space and \(p \in X\), \(\chi(p, X)\) denotes the \textit{character} of \(p\) in \(X\), which is the smallest infinite cardinal \(\kappa\) such that \(p\) has a local base of size not larger than \(\kappa\).NEWLINENEWLINEIt was shown in [\textit{A. V. Arkhangel'skij} and \textit{A. P. Kombarov}, Topology Appl. 35, No. 2--3, 121--126 (1990; Zbl 0707.54018)] that if \(X\) is compact space, \(p \in X\) and \(X^2 \setminus \{(p,p)\}\) is normal, then \(p\) has countable character in \(X\). In terms of the functor \(\mathcal{F}(X) = X^2\), and identifying \(X\) with the diagonal \(\Delta = \{(x,x): x \in X\}\), that result can be restated as follows: if \(p \in X\) and \(\mathcal{F}(X) \setminus \{p\}\) is normal, then \(p\) has countable character in \(X\). In the paper under review, the authors generalize this result by showing that for every seminormal functor \(\mathcal{F}\) of finite degree \(n > 1\) one has the following: if \(p\) is a point of a compact space \(X\) such that \(\chi(p,X) \geqslant \aleph_1\) then \(\mathcal{F}(X)\setminus\{p\}\) is not normal.