The Inverse Limit Property for Subcontinua of Inverse Limits of Set-valued Functions
Keywords:
closed subset theorem, connected imkleinen, inverse limit, inverse limit property, upper semi-continuousAbstract
We examine when a subcontinuum of an inverse limit with set-valued functions is equal to the inverse limit of its projections. There are multiple ways to interpret this problem. We focus on one particular interpretation due to Iztok Banić et al. [Bull. Malays. Math. Sci. Soc. 42 (2017), no. 3, 835-846]. When this does hold, we say that the inverse sequence has $\operatorname{ILP}\left(C\left(\lim f_i\right)\right)$. We identify properties of the bonding functions which imply that the system has $\operatorname{ILP}\left(C\left(\lim _{\llcorner i}\right)\right)$. We also show a relationship between this property and connectedness im kleinen and use this to identify inverse sequences that do not have $\operatorname{ILP}\left(C\left(\lim _{\text {⟵ }}\right)\right)$.
References
Hussam Abobaker and Wlodximierx J. Charatonik, Local connectedness and inverse limits, Questions Answers Gen. Topology 35 (2017), no. 2, 139-141.
Ixtok BaniĆ, Matevž Cörepnjak, Matej Merhar, Uroš Milutinović, and Tina Sović, The closed subset theorem for inverse limits with upper semicontinuous bonding functions, Bull. Malays. Math. Sci. Soc. 42 (2017), no. 3, 835-846.
Wodzimierz J. Charatonik and Faruq A. Mena, Local connectedness of inverse limits, Topology Appl. 265 (2019), 106823. https://doi.org/10.1016/j.topol. 2019.106823
W. T. Ingram, An Introduction to Inverse Limits with Set-valued Functions. Springer Briefs in Mathematics. New York: Springer, 2012.
W. T. Ingram, Concerning chainability of inverse limits on $[0,1]$ with set-valued functions, Topology Proc. 42 (2013), 327-340.
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.
W.T. Ingram and William S. Mahavier, Inverse Limits: From Continua to Chaos. Developments in Mathematics, 25. New York: Springer, 2012.
James P. Kelly, Monotone and weakly confluent set-valued functions and their inverse limits, Topology Appl. 228 (2017), 486-500.
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., Continumm Theory: An Introduction. Monographs and Textbooks in Pure and Applied Mathematics, 158. New York: Marcel Dekker, Inc., 1992.
Van Nall, Inverse limits with set valued functions, Houston J. Math. 37 (2011), no. 4, 1323-1332
Scott Varagona, Inverse limits with upper semi-continuous bonding functions and indecomposability, Houston J. Math. 37 (2011), no. 3, 1017-1034.
