Examples concerning means on dendroids of generalized type N

摘要

Let X be a metric continuum. For each positive integer n, we denote by 2(X) and F-n(X) the hyperspaces of all nonempty closed subsets of X and of all nonempty subsets of X with at most n points. A mean for Xis a retraction r: F-2(X) - F-1(X). It is known that continua of type N do not admit means and it is an open question if this result can be extended to continua (or dendroids) of generalized type N. In 2012, F. Capulin, F. Orozco-Zitli and I. Puga gave an alternative definition of continuum of generalized type N(which we call COP-generalized type N) because apparently the previous definition was inadequate. Each continuum of COP-generalized type N is of generalized type N. In this paper, we present a dendroid of generalized type N admitting a mean. We also prove the non-existence of means for the most natural dendroids of generalized type N with a simple triod as limit; including the most natural dendroid of COP-generalized type N with a simple triod as limit. Finally, we show a contractible, non-selectible planar dendroid X which admits a retraction r: 2(X) - F-1(X) and, in consequence, it admits a mean. (C) 2020 Elsevier B.V. All rights reserved.