Connectedness and inverse limits with set-valued functions on intervals

Inverse component cropping sequences and connected inverse limits over intervals

“…2 [0,1] denotes the set of all nonempty closed subsets of [0,1]. A function f : 1] is upper semicontinuous at a point x ∈…”

“…We say that the graph of a function f : 1] is surjective if for each y ∈ [0, 1] there is a point x ∈ [0, 1] such that y ∈ f (x).…”

“…The following statements are equivalent. As seen in the figure, there is an inverse component cropping sequence over [1,4] for {f…”

“…Also, V. Nall proved the following theorem, giving a characterization of connected inverse limits of inverse sequences of intervals with a single upper semicontinuous bonding function whose graph is surjective ( [7]). [0,1] be an upper semicontinuous function with a surjective graph. The following statements are equivalent.…”

“…This problem has already been solved. The question was answered in the negative (see [1,4]). Motivated by the above problem, we give a characterization of inverse sequences {[0, 1], f i } ∞ i=1 for which the inverse limit lim ← − {[0, 1], f −1 i } ∞ i=1 is connected.…”