On a characterization of 𝑊-sets and the dimension of hyperspaces

Grispolakis, J.; Tymchatyn, E. D.

doi:10.1090/S0002-9939-1987-0891163-6

On a characterization of $W$-sets and the dimension of hyperspaces
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by J. Grispolakis and E. D. Tymchatyn PDF

Proc. Amer. Math. Soc. 100 (1987), 557-563 Request permission

Abstract:

A subcontinuum $A$ of a continuum $X$ is a $W$-set if for each mapping $f:Y \twoheadrightarrow X$ 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.

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