Dendrites with a countable set of end points and universality (Q2839890)

A member \(Z\) of a class of topological spaces is a \textit{universal element} (resp., \textit{minimal element}) for the class if everything in the class embeds in \(Z\) (resp., \(Z\) embeds in everything in the class). This paper is about the possible existence of universal/minimal elements for classes of \textit{dendrites}, locally connected metrizable continua that contain no simple closed curves.NEWLINENEWLINEIf \(x\) is a point of a space \(X\), its \textit{order} in \(X\) is the least cardinal \(\kappa\) such that \(x\) has arbitrarily small open neighborhoods with boundaries of cardinality \(\leq\kappa\). A point of order 1 is called an \textit{end} point of the space, a point of order \(\geq 3\) is called a \textit{ramification} point.NEWLINENEWLINEAs for old positive results, the authors tell us that the class \(\mathcal{F}\) has a universal element if: (1) \(\mathcal{F}\) is the class of all dendrites [\textit{T.~Wazewski}, Ann. de la soc. polon. de math. 2, 49--170 (1924; JFM 50.0373.02)]; (2) \(\mathcal{F}\) is the class of dendrites where all points have order \(\leq n\) (\(n\) a fixed whole number) [\textit{K.~Menger}, Kurventheorie. Leipzig: B. G. Teubner (1932; Zbl 0005.41504; JFM 58.1205.02)]; or (3) \(\mathcal{F}\) is the class of dendrites whose set of end points is closed [\textit{D.~Arévalo} et al., Topology Appl. 115, No. 1, 1--17 (2001; Zbl 0979.54035)].NEWLINENEWLINEOn the other side, \(\mathcal{F}\) does not have a universal element if it comprises those dendrites whose set of end points is both closed and countable [\textit{S.~Zafiridou}, Topology Appl. 155, No. 17--18, 1935--1946 (2008; Zbl 1165.54013)]. In the present paper, the authors introduce the ramification degree for dendrites, and use this ordinal invariant to get non-existence results for universal/minimal elements of various other classes of dendrites.NEWLINENEWLINEFor a dendrite \(X\), define \(X_{(0)}\) to be \(X\); and for \(\alpha\) any ordinal, define \(X_{(\alpha+1)}\) to be the (unique, since \(X\) is hereditarily unicoherent) subcontinuum of \(X_{(\alpha)}\) that is irreducible about its set of ramification points (with the empty set possible). Taking intersections at limit ordinals, the \textit{ramification degree} of \(X\) is the smallest ordinal \(\alpha\) -- known to be countable, if it exists -- such that \(X_{(\alpha)}\) is empty. (The ramification degree is undefined otherwise.)NEWLINENEWLINEFrom the techniques developed in this paper, there is no universal element for \(\mathcal{F}\) if: (1) \(\mathcal{F}\) comprises the dendrites with countably many end points; or (2) \(\mathcal{F}\) comprises the dendrites with countably many end points, and which do not contain a copy of the planar dendrite \(([-1,1]\times\{0\})\cup\bigcup_{n=1}^{\infty}(\{\frac{1}{n}\}\times [0,\frac{1}{n}])\). Also there is no minimal element for \(\mathcal{F}\) if \(\mathcal{F}\) comprises the family of dendrites of ramification degree \(\geq\alpha+1\) (\(\alpha\) a fixed countably infinite ordinal).