AbstractWe study retractions from the hyperspace of all nonempty closed subsets of a given continuum onto the continuum (which is naturally embedded in the hyperspace). Some necessary and some sufficient conditions for the existence of such a retraction are found if the continuum is a curve. It is shown that the existence of such a retraction for a curve implies that the curve is a uniformly arcwise connected dendroid, and that a universal smooth dendroid admits such a retraction. The existence of this retraction for a given dendroid implies that the dendroid admits a mean. An example of a (nonplanable) smooth dendroid that admits no mean is constructed. Some related results are obtained and open problems are stated. The results answer several questions asked in the literature.
CONTENTS1. Introduction...................................52. Preliminaries.................................73. Hyperspace retractions.................94. Applications to selections............155. Applications to means.................18References.....................................32
1991 Mathematics Subject Classification: 54B20, 54C15, 54F15, 54F50.
@book{bwmeta1.element.zamlynska-2f7a5858-d496-4f5a-9849-f7d9aff7f411,
author = {Charatonik Janusz J. and Charatonik W\l odzimierz J. and Omiljanowski Krzysztof and Prajs Janusz R.},
title = {Hyperspace retractions for curves},
series = {GDML\_Books},
publisher = {Instytut Matematyczny Polskiej Akademi Nauk},
address = {Warszawa},
year = {1997},
zbl = {0914.54026},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.zamlynska-2f7a5858-d496-4f5a-9849-f7d9aff7f411}
}
Charatonik Janusz J.; Charatonik Włodzimierz J.; Omiljanowski Krzysztof; Prajs Janusz R. Hyperspace retractions for curves. GDML_Books (1997), http://gdmltest.u-ga.fr/item/bwmeta1.element.zamlynska-2f7a5858-d496-4f5a-9849-f7d9aff7f411/