Hyperspace retractions for curves. (English) Zbl 0914.54026

Charatonik, Janusz J.; Charatonik, Włodzimierz J.; Omiljanowski, Krzysztof; Prajs, Janusz R.

Diss. Math. 370, 34 p. (1997).

The authors of this interesting paper consider several special classes of maps on continua. Among other results they show that a curve which is a retract of its hyperspace of closed subsets (or its hyperspace of all subcontinua) must be a uniformly arcwise connected dendroid. Then they consider the universal smooth dendroid \(Y\) which was constructed by L. Mohler and the reviewer in [Houston J. Math. 14, No. 4, 535-541 (1988; Zbl 0684.54023)] proving that \(Y\) has the property of Kelley, admits an associative retraction \(r:2^Y\to Y\), and admits an associative mean. They give an example of a non-planable smooth dendroid which does not admit a mean thus answering a question of M. Bell and S. Watson [ibid. 22, No. 1, 39-50 (1996; Zbl 0860.54031); see also K. Kawamura and E. D. Tymchatyn, Colloq. Math. 71, No. 1, 97-105 (1996; Zbl 0859.54022)].

Reviewer: J.Nikiel (Beirut)

MSC:

54F15	Continua and generalizations
54C15	Retraction
54B20	Hyperspaces in general topology
54F50	Topological spaces of dimension \(\leq 1\); curves, dendrites

Keywords:

curve; dendrite; smooth dendroid; hyperspace; mean; retraction; selection

Citations:

Zbl 0684.54023; Zbl 0859.54022; Zbl 0860.54031