On freely decomposable maps. (English) Zbl 1241.54018

A mapping \(f\) between metric continua \(X\) and \(Y\) is said to be freely decomposable if whenever \(E\) and \(D\) are proper subcontinua of \(Y\) such that \(Y=E\cup D\), then there exist two proper subcontinua \(A\) and \(B\) of \(X\) such that \(X=A\cup B\), \(A\subset f^{-1}(E)\) and \(B\subset f^{-1}(D)\); and the mapping \(f\) is said to be strongly freely decomposable provided that whenever \(E\) and \(D\) are proper subcontinua of \(Y\) such that \(Y=E\cup D\), it follows that \(f^{-1}(E)\) and \(f^{-1}(D)\) are connected. Freely decomposable and strongly freely decomposable mappings were introduced by G. R. Gordh and C. B. Hughes [Glas. Math., III. Ser. 14 (34), 137–146 (1979; Zbl 0411.54011)] as a generalization of monotone maps with the property that they preserve local connectedness in inverse limits. Strongly decomposable mappings were studied by J. J. Charatonik [Topology Appl. 105, No. 1, 15–29 (2000; Zbl 0951.54031)] under the name of feebly monotone maps. In the paper under review, the authors investigate freely decomposable and strongly freely decomposable maps and their relations with other types of maps as weakly monotone, almost monotone and monotone maps.


54E40 Special maps on metric spaces
54F15 Continua and generalizations
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