The closed subset theorem for inverse limits with upper semicontinuous bonding functions (Q2421409)

Let \(X=\displaystyle{\lim_{\leftarrow}\{X_i,f_i\}}\) be an inverse limit of an inverse sequence \(\{X_i, f_i\}_{i=1}^{\infty}\) of compact metric spaces with continuous single-valued bonding functions. \textit{S. B. Nadler jun.} [Continuum theory. An introduction. New York: Marcel (1992; Zbl 0757.54009)] showed that any closed subset of the space \(X\) is the inverse limit of its projections. This is known as ``the closed subset theorem''. \textit{W. T. Ingram} [An introduction to inverse limits with set-valued functions. Berlin: Springer (2012; Zbl 1257.54033)] with an example showed that this theorem does not hold for inverse limits of inverse sequences of compact metric spaces with upper semicontinuous set-valued bonding functions. In the present paper, the authors give sufficient and necessary conditions on the bonding functions \(f_i: X_{i+1}\longrightarrow X_i\) for the inverse limit \(\displaystyle{\lim_{\leftarrow}\{X_i,f_i\}}\) to have the property that for any closed subset \(A\) of \(\displaystyle{\lim_{\leftarrow}\{X_i,f_i\}}\), \(A\) is the inverse limit of its projections.