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On the structure of tranches in continuously irreducible continua. (English) Zbl 0636.54030

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

MSC:

54F15 Continua and generalizations

Citations:

Zbl 0158.415
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