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Local connectivity and maps onto non-metrizable arcs. (English) Zbl 0889.54021
Three subclasses of the class of all locally connected continua are investigated: \({\mathcal R}_M\) – of rim-metrizable continua, \({\mathcal R}_S\) – of rim-scattered, and \({\mathcal R}_{MN}\) – of monotonically normal ones. The main result of the paper states that if a locally connected continuum \(X\) is in one of these three classes, and if for each pair of points \(a,b\in X\) there exists a continuous onto mapping \(f:X \to[c,d]\) such that \(f(a) =c\), \(f(b)=d\) and \([c,d]\) is a non-metrizable arc, then \(X\) is a Hausdorff continuous image of an ordered continuum. Other properties of the continuum \(X\) that satisfies the above assumptions are shown, e.g., that \(X\) is rim-finite. Some related interesting results are presented.

MSC:
54F15 Continua and generalizations
54C05 Continuous maps
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
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