Near-homeomorphisms on hereditarily indecomposable circle-like continua
The reviewer [Proc. Am. Math. Soc. 91, 147-154 (1984; Zbl 0553.54018)] and \textit{M. Smith} [Proc. Am. Math. Soc. 91, 163-166 (1984; Zbl 0511.54020)] have shown that every continuous self-surjection of the pseudo-arc is a near-homeomorphism. The author presents several conditions on inverse limits of simple closed curves for the inverse limit to be hereditarily indecomposable. He then shows that a continuous self-surjection f of an hereditarily indecomposable circularly chainable continuum is a near-homeomorphism if and only if it is a shape equivalence and if and only if it induces an isomorphism on first Čech cohomology. In particular, if f is monotone then it is a near- homeomorphism. An example is constructed of a degree 2 self-surjection of a pseudo-circle which is not a near homeomorphism.
hereditarily indecomposable circularly chainable continuum
scientific article; zbMATH DE number 4143011