Connected generalized inverse limits over intervals (Q2836007)

\textit{W. S. Mahavier} [Topology Appl. 141, No. 1--3, 225--231 (2004; Zbl 1078.54021)] introduced the study of inverse limits with set valued functions.NEWLINENEWLINEFor the particular case in which all the spaces are equal to the interval \([0,1]\) and we have set valued functions \(f_{n}\), the inverse limit is defined as follows: \(\lim_{\leftarrow} ([0,1], f_{n})\) is the set of sequences \((x_n)_{n\in\mathbb N}\) in the Hilbert cube such that \(x_{n+1}\in f(x_{n})\) for each \(n\).NEWLINENEWLINEIn the cited paper, Mahavier also constructed an example of a set valued function \(f\), having connected graph, such that the inverse limit using only this function is not connected.NEWLINENEWLINESince then, a number of authors have been looking for conditions under which the inverse limit is connected.NEWLINENEWLINEIn the paper under review the authors characterize connectedness of inverse limits, for the particular case of functions defined on the interval, using sequences of upper semi-continuous bonding functions having connected graphs. They offer two characterizations, one is given in terms of a property called a CC-sequence (complicated to describe) and the other one is the following: An inverse limit with a sequence of upper semi-continuous bonding functions (defined on the interval) having connected graphs is not connected if and only if there is a basic open proper subset of the product of the factor spaces that contains a component of the inverse limit.