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

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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J. L. Kelley, Hyperspaces of a continuum, Trans. Amer. Math. Soc. 52 (1942), 22–36. MR 6505, DOI 10.1090/S0002-9947-1942-0006505-8
R. J. Koch, Arcs in partially ordered spaces, Pacific J. Math. 9 (1959), 723–728. MR 108553
K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
K. Kuratowski, Topology. Vol. II, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1968. New edition, revised and augmented; Translated from the French by A. Kirkor. MR 0259835
M. M. McWaters, Arcs, semigroups, and hyperspaces, Canadian J. Math. 20 (1968), 1207–1210. MR 231359, DOI 10.4153/CJM-1968-115-3
Ernest Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152–182. MR 42109, DOI 10.1090/S0002-9947-1951-0042109-4
L. E. Ward Jr., Concerning Koch’s theorem on the existence of arcs, Pacific J. Math. 15 (1965), 347–355. MR 181981
Raymond Louis Wilder, Topology of Manifolds, American Mathematical Society Colloquium Publications, Vol. 32, American Mathematical Society, New York, N. Y., 1949. MR 0029491