Whitney levels in $C_ p(X)$ are ARs
Author:
Mark Lynch
Journal:
Proc. Amer. Math. Soc. 97 (1986), 748-750
MSC:
Primary 54B20; Secondary 54C55, 54C56, 54F45
DOI:
https://doi.org/10.1090/S0002-9939-1986-0846000-1
MathSciNet review:
846000
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Abstract | References | Similar Articles | Additional Information
Abstract: For $X$ a metric continuum, and $p \in X$, we show that the Whitney levels in the relative hyperspace ${C_p}(X) = \{ K \in C(X)|p \in K\}$ are absolute retracts.
- D. W. Curtis, Hyperspaces of Peano continua, Topological structures, II (Proc. Sympos. Topology and Geom., Amsterdam, 1978) Math. Centre Tracts, vol. 115, Math. Centrum, Amsterdam, 1979, pp. 51–65. MR 565825
- J. Dugundji, An extension of Tietze’s theorem, Pacific J. Math. 1 (1951), 353–367. MR 44116
- Carl Eberhart, Intervals of continua which are Hilbert cubes, Proc. Amer. Math. Soc. 68 (1978), no. 2, 220–224. MR 480197, DOI https://doi.org/10.1090/S0002-9939-1978-0480197-6
- C. Eberhart and S. B. Nadler Jr., The dimension of certain hyperspaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 19 (1971), 1027–1034. MR 303513
- Sze-tsen Hu, Theory of retracts, Wayne State University Press, Detroit, 1965. MR 0181977
- J. L. Kelley, Hyperspaces of a continuum, Trans. Amer. Math. Soc. 52 (1942), 22–36. MR 6505, DOI https://doi.org/10.1090/S0002-9947-1942-0006505-8
- J. Krasinkiewicz and Sam B. Nadler Jr., Whitney properties, Fund. Math. 98 (1978), no. 2, 165–180. MR 467691, DOI https://doi.org/10.4064/fm-98-2-165-180
- Sam B. Nadler Jr., Hyperspaces of sets, Marcel Dekker, Inc., New York-Basel, 1978. A text with research questions; Monographs and Textbooks in Pure and Applied Mathematics, Vol. 49. MR 0500811
- Ann Petrus, Whitney maps and Whitney properties of $C(X)$, Topology Proceedings, Vol. I (Conf., Auburn Univ., Auburn, Ala., 1976), Math. Dept., Auburn Univ., Auburn, Ala., 1977, pp. 147–172. MR 0451211
- James T. Rogers Jr., Applications of a Vietoris-Begle theorem for multi-valued maps to the cohomology of hyperspaces, Michigan Math. J. 22 (1975), no. 4, 315–319 (1976). MR 397678
- James T. Rogers Jr., Whitney continua in the hyperspace $C(X)$, Pacific J. Math. 58 (1975), no. 2, 569–584. MR 383372
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Additional Information
Keywords:
Absolute retract,
hyperspace of subcontinua,
Whitney level
Article copyright:
© Copyright 1986
American Mathematical Society
