Subcontinua with degenerate tranches in hereditarily decomposable continua

Abstract:

A hereditarily decomposable, irreducible, metric continuum $M$ admits a mapping $f$ onto $[0,1]$ such that each ${f^{ - 1}}(t)$ is a nowhere dense subcontinuum. The sets ${f^{ - 1}}(t)$ are the tranches of $M$ and ${f^{ - 1}}(t)$ is a tranche of cohesion if $t \in \{ 0,1\}$ or ${f^{ - 1}}(t) = {\text {C1}}({f^{ - 1}}([0,t))) \cap {\text {C1}} ({f^{ - 1}}((t,1]))$. The following answer a question of Mahavier and of E. S. Thomas, Jr. Theorem. Every hereditarily decomposable continuum contains a subcontinuum with a degenerate tranche. Corollary. If in an irreducible hereditarily decomposable continuum each tranche is nondegenerate then some tranche is not a tranche of cohesion. The theorem answers a question of Nadler concerning arcwise accessibility in hyperspaces.