Concerning chainability of inverse limits on \([0,1]\) with set-valued functions (Q2850659)

In this paper the author considers the following problem: find sufficient conditions on an upper-semicontinuous function \(f: [0,1]\to 2^{[0,1]}\) so that \(\varprojlim \mathbf{f}\) is a chainable continuum. It is not known to the author whether there is a chainable continuum other than an arc that can be obtained as an inverse limit with a single set-valued function that is not a mapping. In this paper such an example is constructed. In fact, the author proves among other things the following. Let \(f : [0, 1] \to C([0, 1])\) be the upper semi-continuous function whose graph consists of three straight line intervals lying in \([0, 1]^2\): one from \((0, 0)\) to \((1/2, 1)\), one from \((1/2, 1)\) to \((1/2, 0)\), and one from \((1/2, 0)\) to \((1, 1)\). Then \(\varprojlim \mathbf{f}\) is an indecomposable chainable continuum such that every nondegenerate proper subcontinuum is an arc. The conclusion is that the above problem is a good one.