Topology Proceedings 42 (2013) pp. 327-340: Concerning Chainability of Inverse Limits on $[0,1]$ with Set-Valued Functions

Suppose that X1, X2, X3, . . . is a sequence of continua and fi : Xi+1 → 2Xi is an upper semi-continuous set-valued function for each positive integer i; let Gn = {x ∈ ∏n+1 i=1 Xi | xi ∈ fi(xi+1) for 1 ≤ i ≤ n}. We show that if Gn is a chainable continuum for each n ∈ N, then the inverse limit lim ←− is chainable. We use this to show that two well-studied examples of inverse limits on [0, 1] with set-valued functions, along with two companion examples (the inverse limit with the inverses of those functions), are chainable. In the process we prove a union theorem for chainable continua, specifically, if A and B are chainable continua such that A ∩ B is a terminal C-set in each of A and B, then A ∪ B is chainable.