The depth of tranches in $\lambda$-dendroids

Abstract:

According to the well-known theory of Kuratowski, any hereditarily decomposable chainable continuum admits a decomposition into tranches. These tranches are themselves chainable and thus admit decompositions into their own tranches. We may thus define nested sequences $\{ {T_\alpha }\}$ of tranches-within-tranches, indexed by countable ordinals $\alpha$, and finally terminating in a singleton set. E. S. Thomas, Jr. has asked whether, for a given continuum $C$, there is a countable ordinal bound on the length of all such nests $\{ {T_\alpha }\}$ in $C$. We answer Thomas’s question in the affirmative. By generalizing the definitions, we obtain the same result for $\lambda$-dendroids. We also answer, for chainable continua, a related question of Illiadis.