Cardinal sequences
A topological space \(X\) is called scattered provided that every subspace of \(X\) contains an isolated point. Let \(X\) be a locally compact scattered (LCS) topological space. Let \(I_{0}(X)\) be the set of all isolated points of \(X\), and for an ordinal \(\alpha\) define \(I_{\alpha}(X)\) to be the set of all isolated points of the subspace \(X\setminus\bigcup\{ I_{\beta}(X):\beta<\alpha\}\). The last ordinal \(\alpha\) for which \(I_{\alpha}(X)=\emptyset\) is called the height of \(X\), written \(ht(X)\). Each \(p\in X\) is in a unique \(I_{\alpha}(X)\); this \(\alpha\) is called the height of \(p\) in \(X\) and is written \(ht(p)\). The sequence of cardinals \(| I_{\beta}(X)| \), \(\beta< ht(X)\), is called the cardinal sequence of the space \(X\) and denoted by \(SEQ(X)\). The authors characterize all those sequences of cardinals of length \(\omega_1\) which are cardinal sequences of some LCS space: {A sequence \(\kappa_{\alpha}\), \(\alpha<\omega_1\), is the cardinal sequence of a LCS space iff \(\kappa_{\delta}\leq \kappa_{\xi}^{\omega}\) for each \(\xi<\delta<\omega_1\).} Unlike for sequences of length \(\omega_1\), it is not possible to characterize in ZFC alone those sequences of cardinals of length \(\omega_1+1\) which are the cardinal sequences of LCS spaces. However, the authors formulate a sufficient condition under which a sequence of length \(<\omega_2\) is a cardinal sequence of a LCS space.
scientific article; zbMATH DE number 5121371