A continuum having its hyperspaces not locally contractible at the top

Illanes, Alejandro

doi:10.1090/S0002-9939-1991-1037209-7

A continuum having its hyperspaces not locally contractible at the top
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by Alejandro Illanes

Proc. Amer. Math. Soc. 111 (1991), 1177-1182

DOI: https://doi.org/10.1090/S0002-9939-1991-1037209-7

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Abstract:

For a continuum $X$, let $C(X)$ (resp. ${2^x}$) be the spaces of all nonempty subcontinua (resp. closed subsets) of $X$. In this paper we answer a question of Dilks by showing an example of a continuum $X$ such that if $H = C(X){\text { or }}{2^x}$, then $H$ does not have nonempty open subsets which are contractible in $H$. In particular, $H$ is not locally contractible at any of its points.

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