Conditions for chainability of inverse limits on \([0,1]\) with interval-valued functions (Q6545212)

The paper presents a continuation of the work by \textit{W. T. Ingram} and \textit{M. M. Marsh} [Topol. Proc. 56, 305--320 (2020; Zbl 1447.54029)], where the authors provide sufficient conditions for the inverse limit of an inverse sequence on \([0,1]\) with upper semi-continuous bonding function to be chainable. In this paper, all results are restricted to cases where the bonding functions are interval-valued. Under this assumption, necessary conditions on bonding functions for chainability of inverse limits space are provided. In Section 5, we find a characterization of chainable inverse limits on unit intervals with interval-valued bonding functions. This characterization is based on the properties of the bonding functions and the induced functions from the \({i+1}\)-th coordinate space into the Mahavier product \(G_1^n\).\N\NIn the last section, three examples are established to show effectiveness of the provided technique in detecting chainability of inverse limits. All these examples also show that this tool can be useful for characterizing the homogeneity of inverse limits of inverse sequences with Markov-type set-valued functions.\N\NFinally, let us mention that the author also posed the question if the properties provided for necessity of chainability of inverse limit already characterize the chainability.