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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Arcs in hyperspaces which are not compact


Author: L. E. Ward
Journal: Proc. Amer. Math. Soc. 28 (1971), 254-258
MSC: Primary 54.55
MathSciNet review: 0275376
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0275376-0
PII: S 0002-9939(1971)0275376-0
Keywords: Hyperspace, arc, partially ordered space
Article copyright: © Copyright 1971 American Mathematical Society