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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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