Inverse limits of upper semi-continuous functions, connectedness and the Cantor set (Q2665198)

When we take a sequence of mappings between metric continua \(f_{n}:X_{n+1}\rightarrow X_{n}\), the inverse limit lim\(_{\leftarrow}(X_{n},f_{n})\) is also a continuum. This does not hold in general when we change mappings by upper semi-continuous set-valued (usc) functions since, in this case, the inverse limit can result disconnected. Many authors have studied the problem of giving necessary or sufficient conditions to obtain the connectedness on these limits. The problem is difficult and interesting even for the case that all the continua are equal to a continuum \(X\) and all the upper semi-continuous functions are equal to an usc function \(f\). In this direction, \textit{W. T. Ingram} in [Topol. Proc. 36, 353--373 (2010; Zbl. 1196.54056)], restricted the conditions even more by considering the case that the graph of the usc function \(f\) is the union of graphs of single-valued mappings. In the paper under review, assuming Ingram's restrictions the authors give sufficient conditions so that the inverse limit is either a continuum or a Cantor set. The authors also offer a series of relevant examples concerning this topic.