Results on hyperspaces

A characterization of Cp(X), the family of subcontinua of X containing a fixed point of X, when X is an atriodic continuum is given as follows. Assume Z is a continuum and consider the following three conditions: (1) Z is a planar absolute retract; (2) cut points of Z have component number two; (3) any true cyclic element of Z contains at most two cut points of Z. If X is an atriodic continuum and p ∈ X, then Cp(X) satisfies (1)--(3) and, conversely, if Z satisfies (1)--(3), then there exist an arc-like continuum (hence, atriodic) X and a point p ∈ X such that Cp(X) is homeomorphic to Z. For n ≥ 3, it is shown that the n th symmetric product of nondegenerate continua is mutually aposyndetic, and that the natural map of the Cartesian product onto the n th symmetric product of nondegenerate continua is not k-confluent for any positive integer k. It is also shown that for every nondegenerate continuum X there is a non-k-confluent map of some continuum onto F2(X) for any positive integer k. Answers are provided to questions of S. Macias and S. B. Nadler, Jr., about when the space of singletons is a Z-set in the hyperspace C( X). In answering one of these questions it is shown in general that C(X) being contractible is sufficient, but not necessary, for X to be a C(X)-coselection space