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On the Maćkowiak-Tymchatyn theorem. (English) Zbl 1072.54022
A continuum is a compact connected space, and a continuum is decomposable if it can be written as the union of two of its proper subcontinua. Otherwise we say it is indecomposable. If every subcontinuum of a continuum, \(X\), is indecomposable then we say that \(X\) is hereditarily indecomposable. A continuous mapping \(f:X\to Y\) is called weakly confluent provided that every subcontinuum in the subcontinuum of \(Y\) is an image of some subcontinuum of \(X\). The Maćkowiak and Tymchatyn theorem states that every metric continuum is a weakly confluent image of some one-dimensional hereditarily indecomposable metric continuum. In the paper under review the authors utilize model-theoretic techniques to prove the Maćkowiak and Tymchatyn theorem. This is a quite interesting application of model theory to metric continuum theory.
MSC:
54F15 Continua and generalizations
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
06D05 Structure and representation theory of distributive lattices
03C98 Applications of model theory
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