A theorem on decomposition into compact-scattered subspaces and cardinality of topological spaces (Q1329403)

The main result of the paper under review says that if in a space \(X\) each subspace is the union of no more than \(\lambda\) compact sets, then \(| X | \leq \lambda^ \omega\). Recently \textit{J. Gerlits}, \textit{A. Hajnal} and \textit{Z. Szentmiklóssy} [Discrete Math. 108, No. 1-3, 31-35 (1992; Zbl 0766.54002)] have shown that if moreover \(X\) is Hausdorff, then \(| X|\leq\lambda\). Both theorems give partial answers to a question of A. V. Arkhangel'skij. The connections between this problem and the problem of decomposability of a topological space into two spaces containing no non-scattered compact sets are discussed as well; see also [the authors, Čas. Pěstování Mat. 109, 27-53 (1984; Zbl 0539.54005)].