On simplicial maps and chainable continua
A continuum is said to be of surjective span \(\geq {\mathfrak d}> 0\) if it admits parametrizations by a parameter running over a continuum such that the distance between the values of parametrizations is at each value of the parameter not less than \({\mathfrak d}\). The extremal case of continua of surjective span zero is of special interest, and the conjecture that continua of surjective span zero are chainable is to the present day not confirmed. The difficulty of the problem is illustrated by the paper under review in which the author presents a partial solution. As the problem reduces to tree-like continua -- even atriodic ones [\textit{H. Kato}, \textit{A. Koyama}, and \textit{E. D. Tymchatyn}, Houston J. Math. 17, 325-333 (1991; Zbl 0765.54025)], the author concentrates his considerations on inverse sequences of (finite) trees and proves that if a continuum of surjective span zero is represented as the inverse limit of trees with simplicial bonding maps then it is chainable. The proof consists in elaborating a method of factoring simplicial maps between trees through arcs. To do this, a method of tree-words due to \textit{L. Mohler} and \textit{L. G. Oversteegen} [Fundam. Math. 126, 107-121 (1986; Zbl 0619.05044)] is applied.
scientific article; zbMATH DE number 567697
scientific article