Connected generalized inverse limits
For \(i>0\), let \(f_i:I_i\to 2^{I_{i-1}}\) be a surjective upper semicontinuous function whose graph is connected, where the \(I_i\) are closed intervals. The authors introduce a technical condition, called a C-sequence, on the sequence \(\{f_i\}\), which implies that the inverse limit of this sequence is not connected. More precisely, they prove that the sequence \(\{f_i\}\) has a C-sequence iff there exist integers \(n\geq0\) and \(m>n+1\) such that, for \(n\leq i\leq m\), there exist an open interval \(U_i\) and a closed interval \(A_i\) with \(A_i\subset U_i\subset I_i\), \(U_i\neq I_i\) and \(\varprojlim(I_i,f_i)\cap(\prod_{0\leq i<n}I_i \times \prod_{n\leq i\leq m}U_i\times\prod_{i=m+1}^\infty I_i)=\varprojlim(I_i,f_i)\cap(\prod_{0\leq i<n}I_i\times\prod_{n\leq i\leq m}A_i \times\prod_{i=m+1}^\infty I_i)\neq\emptyset\).
scientific article; zbMATH DE number 5986541
scientific article