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On a characterization of \(W\)-sets and the dimension of hyperspaces. (English) Zbl 0618.54010

A subcontinuum \(A\) of a continuum \(X\) is a \(W\)-set if for each mapping \(f\colon Y \twoheadrightarrow X\) of an arbitrary continuum \(Y\) onto \(X\) there is a continuum in \(Y\) which is mapped by \(f\) onto \(A\). We characterize \(W\)-sets in terms of accessibility by small continua. We localize several known results on continua all of whose subcontinua are \(W\)-sets. Finally, we extend a result of J. T. Rogers by proving that if \(X\) is an atriodic continuum whose first Čech cohomology group is finitely generated then the hyperspace \(C(X)\) of subcontinua of \(X\) is two dimensional.
Reviewer: J. Grispolakis

MSC:

54B20 Hyperspaces in general topology
54F15 Continua and generalizations
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
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