Mohler, Lee; Oversteegen, Lex G. On the structure of tranches in continuously irreducible continua. (English) Zbl 0636.54030 Colloq. Math. 54, No. 1, 23-28 (1987). A continuum X is said to be irreducible if there are points p,q\(\in X\) such that no proper subcontinuum of X contains both p and q. If X is irreducible and every indecomposable subcontinuum of X has void interior, then there is a monotone mapping \(m: X\to I=[0,1]\) such that every other monotone mapping of X onto I can be factorized through m [K. Kuratowski, Topology. II (Warszawa, 1968), p. 200]. \(m^{-1}(t)\), \(t\in I\), are called tranches or layers of X. If continuum-valued map \(m^{- 1}\) is continuous, then X is said to be continuously irreducible. In 1957, Knaster asked whether every continuously irreducible continuum must contain a hereditarily indecomposable tranche. Using the notion of atomic maps [cf. H. Cook, Fundam. Math. 60, 241-249 (1967; Zbl 0158.415)], the authors give counterexamples to Knaster’s question. Reviewer: A.Asada Cited in 1 ReviewCited in 2 Documents MSC: 54F15 Continua and generalizations Keywords:monotone mapping; continuously irreducible continuum; hereditarily indecomposable tranche; atomic maps Citations:Zbl 0158.415 PDFBibTeX XMLCite \textit{L. Mohler} and \textit{L. G. Oversteegen}, Colloq. Math. 54, No. 1, 23--28 (1987; Zbl 0636.54030) Full Text: DOI