Connected generalized inverse limits over intervals

Sina Greenwood ¹   ; Judy Kennedy ²

¹ University of Auckland Private Bag 92019 Auckland, New Zealand
² Department of Mathematics Lamar University P.O. Box 10047 Beaumont, TX 77710, U.S.A.

Fundamenta Mathematicae, Tome 236 (2017) no. 1, pp. 1-43

Résumé

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 $\underleftarrow{\lim}\,(I_i,f_i)$.

Détail

DOI : 10.4064/fm241-4-2016

Keywords: suppose each geq closed interval rightarrow surjective upper semicontinuous function connected graph condition graphs called cc sequence underleftarrow lim i disconnected only system admits cc sequence underleftarrow lim i disconnected only there basic proper subset prod contains component underleftarrow lim i