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Decompositions of continua over the hyperbolic plane. (English) Zbl 0704.54020

Summary: The following theorem is proved: Let X be a homogeneous continuum such that \(H^ 1(X)\neq 0\). If \({\mathcal G}\) is the collection of maximal terminal proper subcontinua of X, then (1) The collection \({\mathcal G}\) is a monotone, continuous, terminal decomposition of X, (2) The nondegenerate elements of \({\mathcal G}\) are mutually homeomorphic, indecomposable, cell- like, terminal, homogeneous continua of the same dimension as X, (3) The quotient space is a homogeneous continuum, and (4) The quotient space does not contain any proper, nondegenerate, terminal subcontinuum.
This theorem is related to the Jones’ Aposyndetic Decomposition Theorem [F. B. Jones, Topology, Proc. Cont., Houston/Tex. 1983, Topology Proc. 8, No.1, 51-54 (1983; Zbl 0537.54022)]. The proof involves the hyperbolic plane and a subset of the circle at \(\infty\), called the set of ends of a component of the universal cover of X.

MSC:

54F15 Continua and generalizations
54F50 Topological spaces of dimension \(\leq 1\); curves, dendrites

Citations:

Zbl 0537.54022
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References:

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