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Continua determined by mappings. (English) Zbl 0946.54025

A family \({\mathcal C}\) of compact, connected, metric spaces (continua) is determined by a class \({\mathcal M}\) of mappings if a necessary and sufficient condition for \(Y\in {\mathcal C}\) is that every epimorphic mapping \(f : Z\rightarrow Y\) is an element of \({\mathcal M}\) for any continuum \(Z\). The authors characterize various classes \({\mathcal C}\) of continua by describing the associated determining families \({\mathcal M}_{\mathcal C}\). For example they show that \(Y\) is indecomposable iff \(Y\) is determined by “almost monotone” mappings and that \(Y\) is hereditarily indecomposable iff \(Y\) is determined by the class of so-called confluent mappings. Along similar lines they show that \(Y\) does not contain any subcontinuum which is an union of a \(3\)-chain iff \(Y\) is determined by the class of so-called joining mappings. They systematically analyze results of this type and among other classes of maps arising this way are so-called atriodic mappings, universal mappings etc.

MSC:

54F15 Continua and generalizations
54E40 Special maps on metric spaces
54F65 Topological characterizations of particular spaces
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