Some structure theorems for inverse limits with set-valued functions (Q2850653)

In this paper, some properties of inverse limits with set-valued bonding functions are investigated. Let \(\{ X_i\}\) be a sequence of compact metric spaces. A sequence \(\{ X_i , G_i^{i+1}\}\) is said to be an inverse sequence with set-valued bonding functions if \(G_i^{i+1} : X_{i+1} \to X_i\) is an upper semicontinuous set-valued function for each \(i \geq 1\). The inverse limit \(\lim\limits_{\longleftarrow} \{X_i, G_i^{i+1}\}\) is the subspace \(\{ (x_1, x_2 , \dots ) \in \prod _{i\geq 1 } X_i \mid x_i \in G_i^{i+1} (x_{i+1}) \text{ for } n \geq 1 \}\) of the product space \(\prod _{n\geq 1 } X_n \).NEWLINENEWLINEConcerning the connectedness of inverse limits, the author proves the following theorem: Let \(\{X_i\}\) be a sequence of continua (i.e.\ compact connected metric spaces) and suppose that for each \(i \geq 1\), \(G_i^{i+1} : X_{i+1} \to X_i\) is a surjective upper semicontinuous set-valued function whose inverse graph \(\{ (x_{i}, x_{i+1} ) \in X_{i} \times X_{i+1} \mid x_i \in G_i^{i+1} (x_{i+1}) \}\) is connected. Suppose also that for each \(i\geq 1\), \(G_i^{i+1}\) is a union of upper semicontinuous continuum-valued functions. Then \(\lim\limits_{\longleftarrow} \{X_i, G_i^{i+1}\}\) is a continuum. This is a generalization of theorems obtained by \textit{W. T. Ingram} and \textit{W. S. Mahavier} [Houston J. Math.\ 32, No. 1, 119--130 (2006; Zbl 1101.54015)] and \textit{Van Nall} [Topol.\ Proc.\ 40, 167--177 (2012; Zbl 1261.54023)].NEWLINENEWLINEFor an inverse sequence \(\{ X_i , G_i^{i+1}\}\) and a positive integer \(k\), a sequence \(\{Z_i^{i+1} : Y_{i+1} \to X_i\}_{i\geq k}\) of upper semicontinuous set-valued functions is called a \(k\)-tail sequence if for each \(i\geq k\), \(Y_{i+1}\) is a compact subspace of the range of \(Z_{i+1}^{i+2}\), the graph \(\{ (x_{i+1}, x_i ) \in Y_{i+1} \times X_i \mid x_i \in Z_i^{i+1} (x_{i+1}) \}\) of \(Z_i^{i+1}\) is contained in the graph of \(G_i^{i+1}\), and \( \{ y \in Y_{i+1} \mid x \in Z_i^{i+1} (y) \}\) consists of at most one point for each \(x \in X_i\). Using this concept, the author proves the following theorem: Let \(X=\lim\limits_{\longleftarrow} \{X_i, G_i^{i+1}\}\), where for each \(n\geq 1\), \(X_n\) is a compactum and \(G_n^{n+1} : X_{n+1} \to X_n\) is an upper semicontinuous set-valued function. Suppose that there exists a \(k\)-tail sequence \(\{Z_i^{i+1}\}_{i\geq k}\) for some \(k\) such that the range of \(Z_i^{i+1}\) is equal to \(X_i\) for each \(i \geq k\). Then \(X\) can be realized as the limit of some ordinal inverse sequence \(\{A_i, r_i\}_{i\geq k}\), where \(A_n\) is homeomorphic to the inverse graph \(\overleftarrow{\text{gr}} \, G_1^n=\{x \in \prod_{i=1}^n \mid x_i \in G_{i}^{i+1} (x_{i+1}) \text{ for } 1\leq i < n\}\) and \(r_n: A_{n+1}\to A_n\) is a retraction. If, moreover, all \(X_n\) and \(X\) are connected and each inverse graph \(\overleftarrow{\text{gr}}\,G_1^n\) has the fixed point property, then \(X\) has the fixed point property.NEWLINENEWLINEInverse limits on one compactum with a single set-valued bonding functions are also investigated.