A continuum is a compact connected metric space. For a continuum X, let C(X) denote the hyperspace of subcontinua of X. In this paper we construct two nonhomeomorphic fans (dendroids with only one ramification point) X and Y such that C(X) and C(Y) are homeomorphic. This answers a question by Sam B. Nadler, Jr.
@article{bwmeta1.element.bwnjournal-article-cmv81i2p299bwm,
author = {Alejandro Illanes},
title = {Fans are not c-determined},
journal = {Colloquium Mathematicae},
volume = {79},
year = {1999},
pages = {299-308},
zbl = {0965.54014},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv81i2p299bwm}
}
Illanes, Alejandro. Fans are not c-determined. Colloquium Mathematicae, Tome 79 (1999) pp. 299-308. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv81i2p299bwm/
[000] [1] G. Acosta, Hyperspaces with unique hyperspace, preprint. | Zbl 1046.54024
[001] [2] R. D. Anderson, Topological properties of the Hilbert cube and the infinite product of open intervals, Trans. Amer. Math. Soc. 126 (1967), 200-216. | Zbl 0152.12601
[002] [3] R. Duda, On the hyperspace of subcontinua of a finite graph, I, Fund. Math. 69 (1968), 265-286. | Zbl 0167.51401
[003] [4] C. Eberhart and S. B. Nadler, Jr., Hyperspaces of cones and fans, Proc. Amer. Math. Soc. 77 (1979), 279-288. | Zbl 0413.57010
[004] [5] A. Illanes, Chainable continua are not C-determined, Topology Appl., to appear. | Zbl 0964.54004
[005] [6] J. Krasinkiewicz, On the hyperspaces of snake-like and circle-like continua, Fund. Math. 83 (1974), 155-164. | Zbl 0271.54024
[006] [7] S. Macías, On C-determined continua, Glas. Mat., to appear.
[007] [8] S. Macías, Hereditarily indecomposable continua have unique hyperspace , preprint. | Zbl 0938.54010
[008] [9] S. B. Nadler, Jr., Hyperspaces of Sets, Monographs Textbooks Pure Appl. Math. 49, Marcel Dekker, New York, 1978.
[009] [10] L. E. Ward, Extending Whitney maps, Pacific J. Math. 93 (1981), 465-469. | Zbl 0457.54008