Big and large continua in inverse limits of inverse systems over directed graphs (Q2307605)

Ingram and Mahavier [\textit{W. T. Ingram} and \textit{W. S. Mahavier}, Houst. J. Math. 32, 119--130 (2006; Zbl 1101.54015); \textit{W. T. Ingram}, An introduction to inverse limits with set-valued functions. Berlin: Springer (2012; Zbl 1257.54033)] generalized inverse limits of inverse sequences with continuous bonding functions to inverse limits of inverse sequences with upper semicontinuous bonding functions. It is well known that inverse limits of inverse sequences of continua with continuous bonding functions are always connected while generalized inverse limits of inverse sequences of continua may not be [\textit{W. T. Ingram} and \textit{W. S. Mahavier}, loc. cit., Example 1]. ``The notion of generalized inverse limits on intervals has been generalized in another direction as well. Instead of using positive integers as the index set, several authors have studied generalized inverse limits with different index sets, see [\textit{I. Banič} and \textit{T. Sovič}, Bull. Aust. Math. Soc. 89, 49--59 (2014; Zbl 1291.54028)], where more references can be found. It turns out that some continua cannot be obtained as generalized inverse limits of inverse sequences of intervals (indexed by the positive integers) with a single upper semicontinuous bonding function while they can be obtained as generalized inverse limits of inverse systems of intervals indexed by the integers with a single upper semicontinuous bonding function.'' In the present paper, the authors generalize the notion of generalized inverse limits to inverse limits of inverse systems over directed graphs. Using directed graphs, they generalize generalized inverse limits over (positive) integers and show that under certain conditions, such inverse limits also contain large continua. Then, they characterize inverse systems for which every large continuum in the inverse limit is also a big continuum and vice versa. Finally, the authors give several open problems.