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Semi-confluent mappings. (English) Zbl 1062.54016

From the introduction: Let \(f:X\longrightarrow Y\) be a mapping between topological spaces. Then \(f\) is said to be: confluent provided that for each subcontinuum \(Q\) of \(Y\) and for each component \(C\) of \(f^{-1}(Q)\) we have \(f(C)=Q\); weakly confluent provided that for each subcontinuum \(Q\) of \(Y\) there is a component \(C\) of \(f^{-1}(Q)\) for which we have \(f(C)=Q\); and semi-confluent provided that for each subcontinuum \(Q\) of \(Y\) and for every two components \(C_{1}\) and \(C_{2}\) of \(f^{-1}(Q)\) we have \(f(C_{1})\subset f(C_{2})\) or \(f(C_{2})\subset f(C_{1})\).
The paper consists of four sections. In the first of them conditions are discussed which imply that the composition of semi-confluent mappings is semi-confluent. The second section is devoted to the localization of the global concept of semi-confluence: pointed versions of semi-confluent mappings are introduced and studied. Inverse limits are studied in the third section. First, it is shown that if a class of mappings between compact spaces has the inverse limit property, then it has the inverse limit projection property. Next, the inverse limit property is obtained for pointed versions of semi-confluent and strongly semi-confluent mappings. As a consequence it is shown that the class of semi-confluent mappings has the inverse limit property and the inverse limit projection property.
Finally, two examples related to hereditarily indecomposable continua are presented in Section 4. The first of them shows that the known characterizations of these continua in terms of confluent and semi-confluent mappings cannot be extended to wider classes of mappings (for instance, to joining mappings). The second example is related to the Eilenberg property. It indicates that the presence of hereditarily indecomposable continua is not necessary (in both domain and range spaces) for examples showing that the classes of mappings larger than the ones of semi-confluent mappings (such as locally semi-confluent, weakly confluent, or joining confluent mappings) do not have the Eilenberg property.
The paper also contains several open problems related to the obtained results.

MSC:

54C10 Special maps on topological spaces (open, closed, perfect, etc.)
54E40 Special maps on metric spaces
54F15 Continua and generalizations
54B10 Product spaces in general topology
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