\(R^ i\)-continua and hyperspaces (Q1083710)

Following the terminology of the author we call a singularity of metric continua singularity R if the continuum contains a certain kind of continua as introduced and studied by \textit{S. T. Czuba} [Bull. Acad. Pol. Sci., Sér. Sci. Math. 27, 299-302 (1979; Zbl 0424.54026)]. Singularity R, if possessed by a continuum, implies its non-contractibility. The singularity R is preserved by passing from X to the hyperspace \(2^ X\), and under some conditions, by passing to the hyperspace C(X). However, there are continua X with non-contractible \(2^ X\) and not having the singularity R. Also the following theorem is proved: if a continuum X has singularity R, then there does not exist a confluent Whitney map for \(2^ X\).