Connected generalized inverse limits over intervals
Volume 236 / 2017
Fundamenta Mathematicae 236 (2017), 1-43
MSC: 54C08, 54E45, 54F15, 54F65.
DOI: 10.4064/fm241-4-2016
Published online: 4 August 2016
Abstract
Suppose that for each $i\geq 0$, $I_{i}$ is a closed interval and $f_{i+1}:I_{i+1}\rightarrow 2^{I_{i}}$ is a surjective upper semicontinuous function with a connected graph. We give a condition on the graphs called a CC-sequence, and show that $\underleftarrow{\lim}\,(I_i,f_i)$ is disconnected if and only if the system admits a CC-sequence. We also show that $\underleftarrow{\lim}\,(I_i,f_i)$ is disconnected if and only if there is a basic open proper subset of $\prod_{i\ge 0}I_i$ that contains a component of $\underleftarrow{\lim}\,(I_i,f_i)$.
