Connectedness of inverse limits with functions \(f_{i}\) where either \(f_{i}\) or \(f_i^{- 1}\) is a union of continuum-valued functions (Q2315335)

In the Hausdorff setting, \textit{W. T. Ingram} and \textit{W. S. Mahavier} [Houston J. Math. 32, No. 1, 119--130 (2006; Zbl 1101.54015)] proved that inverse limits on continua, where all bonding functions are continuum-valued or the inverses of all bonding functions are continuum-valued, are connected. Also in the Hausdorff setting, \textit{W. T. Ingram} [Topol. Proc. 36, 353--373 (2010; Zbl 1196.54056)] proved that if \(X\) is a continuum and \(f : X \longrightarrow 2^X\) is a set-valued function with a closed graph that is the union of continuum-valued functions, one of which is surjective and universal with respect to the others, then \(\varprojlim\{X,f\}\) is connected. In [ibid. 40, 167--177 (2012; Zbl 1261.54023)], \textit{V. Nall} proved that if \(X\) is a continuum and \(f : X \longrightarrow 2^X\) is a union of continuum-valued functions, then \(\varprojlim\{X,f\}\) is connected. In [ibid. 42, 237--258 (2013; Zbl 1278.54013)], the author generalized Nall's Theorem, in the metric setting, to inverse limits on inverse sequences of continua \(X_i\), and bonding functions where either all \(f_i\) or all \(f_i^{-1}\) are unions of continuum-valued functions. In the present paper, in the setting of inverse sequences \(\{X_i, f_i\}_{i\geq 1}\) on metric continua with surjective upper semi-continuous set-valued bonding functions, where, for each \(i \geq 1\), \(f_i\) has a connected graph, and either \(f_i\) or \(f^{-1}_i\) is a union of continuum-valued functions, the author proves several theorems for connectedness of the partial graphs \(G'(f_1,\dots, f_n)\), and of the inverse limit \(X=\varprojlim\{X_i,f_i\}\). Properties of certain set-valued functions from the factor spaces onto partial graphs in the inverse sequence imply connectedness of \(X\).