Planar rim-scattered spaces and universality (Q2710543)

For every space \(A\) and ordinal \(\alpha\), we define the \(\alpha\)-derivative of \(A\) as follows. Let \(A^{(0)}= A\), let \(A^{(1)}\) be the set of limit points of \(A\), and for an ordinal \(\alpha\), let \(A^{(\alpha)}= (A^{(\alpha- 1)})^{(1)}\) if \(\alpha\) is a successor ordinal, and \(A^{(\alpha)}= \bigcap\{A^{(\beta)}: \beta< \alpha\}\) if \(\alpha\) is a limit ordinal. A space is planar if there exists a homeomorphism of the space onto a subset of the plane. A space \(T\) is universal for a family \({\mathcal A}\) of spaces if both the following conditions are satisfied: (1) \(T\in{\mathcal A}\) and (2) for every \(X\in{\mathcal A}\), there exists a homeomorphism of \(X\) onto a subset of \(T\).NEWLINENEWLINENEWLINEFrom the author's summary: ``We prove that for every limit ordinal \(\alpha> 0\) there is no universal element in the class of all planar spaces (respectively, of all spaces) \(X\) having the following property: for each pair \((F_1, F_2)\) of disjoint closed subsets of \(X\) there exists an open set \(G\subseteq X\) such that \(F_1\subseteq G\subseteq\text{Cl}(G)\subseteq X\smallsetminus F_2\) and the \(\alpha\)-derivative of \(\text{Bd}(G)\) is empty. This gives a partial answer to a problem of \textit{S. D. Iliadis} and the author [Colloq. Math. Soc. János, Bolyai 55, 321-347 (1993; Zbl 0792.54035)]''.