Arcs in hyperspaces which are not compact

Ward, L. E.

doi:10.1090/S0002-9939-1971-0275376-0

Arcs in hyperspaces which are not compact
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by L. E. Ward

Proc. Amer. Math. Soc. 28 (1971), 254-258

DOI: https://doi.org/10.1090/S0002-9939-1971-0275376-0

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Abstract:

It has been known for many years that if $X$ is a metrizable continuum then ${2^X}$ (the space of closed subsets of $X$) and $C(X)$ (the subspace of connected members of ${2^X}$) are arcwise connected. These results are due to Borsuk and Mazurkiewicz [l] and J. L. Kelley [2], respectively. Quite recently M. M. McWaters [6] extended these theorems to the case of continua which are not necessarily metrizable, using Koch’s arc theorem for partially ordered spaces [3], [8]. In this note we prove these results for certain noncompact spaces by means of a simple generalization of Koch’s arc theorem.

References

Sur l’hyperspace d’un continu

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