## 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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