Universal dendrites for some families of dendrites with a countable set of end points
A continuum is a nonempty compact connected space. A continuum \(X\) is called a dendrite if it is locally connected and contains no simple closed curve. The order of a point \(x\) in a space \(X\) is the least cardinal or ordinal number \(\kappa\) such that \(x\) has an arbitrarily small neighborhood in \(X\) with boundary of cardinality \(\leq \kappa\). By order of a dendrite \(X\) we refer to the least cardinal number \(\kappa\) such that the order of every point in \(X\) is less than or equal to \(\kappa\). A point \(x\) of a dendrite \(X\) is called an end point of \(X\) if it has order 1. A space \(Z\) is said to be universal in a class \(\mathcal F\) of spaces provided that both of the following conditions are satisfied: (a) \(Z\in\mathcal F\) and (b) for each \(X\in\mathcal F\) there exists an embedding \(h:X\rightarrow Z\). A classical result due to Waźewski says that there exists a universal dendrite and that for each integer \(\kappa\geq 3\) there exists a universal dendrite in the family of all dendrites with orders of points \(\leq \kappa\). \textit{D. Arévalo, W.J. Charatonik, P. Pellicer Covarrubias} and \textit{L. Simon} constructed a universal element for the family of all dendrites with a closed set of end points. Also for a fixed integer \(\kappa\geq 3\) they constructed a universal element for the family of all dendrites with a closed set of end points and with orders of points \(\leq \kappa\) [Topology Appl. 115, No. 1, 1--17 (2001; Zbl 0979.54035)]. In the present paper, the author proves that for any countable ordinal \(\alpha\geq 1\) (and for any integer \(\kappa\geq 3\)), there exits a universal dendrite in the family of all dendrites (of order \(\leq\kappa\)) with a closed countable set of end points such that the \(\alpha\)-derivative of the set of end points contains at most one point.
scientific article; zbMATH DE number 5365206