Nonconnectedness of inverse limit sequences (Q2850631)

This work deals with inverse sequences \(\mathbf{f}=(f_n)\), \(f_n:X_{n+1}\to 2^{X_n}\) of upper semi-continuous set-valued functions and their limits \(\lim\mathbf{f}\) [\textit{W. T. Ingram}, An introduction to inverse limits with set-valued functions. Berlin: Springer (2012; Zbl 1257.54033)]. The target of this investigation involves the situation in which the bonding maps are such that \(X_n\) is a fixed space \(X\) for all \(n\), but the maps \(f_n\) need not all be the same. In this setting one has \(f_n:X\to 2^X\), and this induces \(f_n^{-1}:X\to 2^X\), and in turn the inverse sequence \(\mathbf{f}^{-1}=(f_n^{-1})\) of upper semi-continuous set-valued functions.NEWLINENEWLINEIn [\textit{V. Nall}, Topol. Proc. 40, 167--177 (2012; Zbl 1261.54023)], it is proved that if \(X\) is a Hausdorff continuum and \(f:X\to 2^X\) is surjective and upper semi-continuous, then \(\lim\mathbf{f}\) is connected if and only if \(\lim\mathbf{f}^{-1}\) is connected. The main result of this paper is to give an example (using \(X=[0,1]\)) showing that without the surjectivity, the just stated result need not hold true. Their example shows that one of the limits is connected, but the other is not.