Connected Generalized Inverse Limits and Intermediate Value Property
Keywords:
connected, generalized inverse limit, intermediate value property, set-valued functionAbstract
In this paper, we consider inverse limits of $[0,1]$ using upper semicontinuous set-valued functions. We introduce two generalizations of the intermediate value property and prove that inverse limits with upper semicontinuous set-valued bonding functions are connected if the bonding functions are surjective, have connected graphs, and have either generalization of the intermediate value property. Examples are given to demonstrate that if any of the conditions is dropped, the result does not hold in general. An example is given to show that an inverse limit may be connected even if the bonding functions do not have either intermediate value property. Further, we compare the structures of set-valued functions with the two types of the intermediate value property.
References
Wlodzimierz J. Charatonik and Robert P. Roe, On Mahavier products, Topology Appl. 166 (2014), 92-97.
Tavish J. Dunn and David J. Ryden, Periodicity and indecomposability in set-valued inverse limits with intermediate value property. Preprint available at https://arxiv.org/abs/2108.04959.
Sina Greenwood and Judy Kennedy, Connected generalized inverse limits, Topology Appl. 159 (2012), no. 1, 57-68.
Sina Greenwood and Judy Kennedy, Connected generalized inverse limits over intervals, Fund. Math. 236 (2017), no. 1, 1-43.
W. T. Ingram, An Introduction to Inverse Limits with Set-valued Functions. Springer Briefs in Mathematics. New York: Springer, 2012.
W. T. Ingram and William S. Mahavier, Inverse limits of upper semi-continuous set valued functions, Houston J. Math. 32 (2006), no. 1, 119-130.
Sergio Macias, Topics on Continua. Boca Raton, FL: Chapman & Hall/CRC, 2005.
William S. Mahavier, Inverse limits with subsets of $[0,1] \times[0,1]$, Topology Appl. 141 (2004), no. 1-3, 225-231.
Sam B. Nadler, Jr. Continuum Theory: An Introduction. Monographs and Textbooks in Pure and Applied Mathematics, 158. New York: Marcel Dekker, Inc., New York, 1992.
Van Nall, Connected inverse limits with a set-valued function, Topology Proc. 40 (2012), 167-177.
Andrew Otey and David J. Ryden, Sharkovskii order for upper semicontinuous functions on $[0,1]$ with Intermediate Value Property. In progress.
