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Whitney levels in hyperspaces of certain Peano continua


Authors: Jack T. Goodykoontz and Sam B. Nadler
Journal: Trans. Amer. Math. Soc. 274 (1982), 671-694
MSC: Primary 54B20; Secondary 54C99, 54F20
DOI: https://doi.org/10.1090/S0002-9947-1982-0675074-7
MathSciNet review: 675074
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Abstract: Let $ X$ be a Peano continuum. Let $ {2^x}$ (resp., $ C(X)$) be the space of all nonempty compacta (resp., subcontinua) of $ X$ with the Hausdorff matric. Let $ \omega$ be a Whitney map defined on $ \mathcal{H}={2^{X}}$ or $ C(X)$ such that $ \omega$ is admissible (this requires the existence of a certain type of deformation of $ \mathcal{H}$). If $ \mathcal{H}=C(X)$, assume $ X$ contains no free arc. Then, for any $ {t_0} \in (0,\omega (X))$, it is proved that $ {\omega ^{ - 1}}({t_0}),\,{\omega ^{ - 1}}([0,\,{t_0}])$, and $ {\omega ^{ - 1}}([{t_0},\,\omega (X)])$ are Hilbert cubes. This is an analogue of the Curtis-Schori theorem for $ \mathcal{H}$. A general result for the existance of admissible Whitney maps is proved which implies that these maps exist when $ X$ is starshaped in a Banach space or when $ X$ is a dendrite. Using these results it is shown, for example that being an AR, an ANR, an LC space, or an $ {\text{L}}{{\text{C}}^n}$ space is not strongly Whitney-reversible.


References [Enhancements On Off] (What's this?)

  • [1] R. H. Bing, Partitioning a set, Bull. Amer. Math. Soc. 55 (1949), 1101-1110. MR 0035429 (11:733i)
  • [2] K. Borsuk, Theory of retracts, Monogr. Mat. 44 (1967). MR 0216473 (35:7306)
  • [3] C. E. Capel and W. L. Strother, Multi-valued functions and partial order, Portugal. Math. 17 (1958), 41-47. MR 0101512 (21:322)
  • [4] J. H. Carruth, A note on partially ordered compacta, Pacific J. Math. 24 (1968), 229-231. MR 0222852 (36:5902)
  • [5] T. A. Chapman, Lectures on Hilbert cube manifolds, CBMS Regional Conf. Ser. in Math., no. 28, Amer. Math. Soc., Providence, R.I., 1975. MR 0423357 (54:11336)
  • [6] D. W. Curtis, Growth hyperspaces of Peano continua, Trans. Amer. Math. Soc. 238 (1978), 271-283. MR 482919 (80a:54009)
  • [7] D. W. Curtis and R. M. Schori, Hyperspaces of Peano continua are Hilbert cubes, Fund. Math. 101 (1978), 19-38. MR 512241 (80k:54010)
  • [8] -, Hyperspaces which characterize simple homotopy type, General Topology and Appl. 6 (1976), 153-165. MR 0394684 (52:15483)
  • [9] Carl Eberhart, Intervals of continua which are Hilbert cubes, Proc. Amer. Math. Soc. 68 (1978), 220-224. MR 480197 (80e:54042)
  • [10] Carl Eberhart and Sam B. Nadler, Jr. Hyperspaces of cones and fans, Proc. Amer. Math. Soc. 77 (1979), 279-288. MR 542098 (80i:54009)
  • [11] -, The dimension of certain hyperspaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. 19 (1971), 1027-1034. MR 0303513 (46:2650)
  • [12] J. B. Fugate, G. R. Gordh, Jr. and Lewis Lum, Arc-smooth continua, Trans. Amer. Math. Soc. 265 (1981), 545-561. MR 610965 (82j:54072)
  • [13] John L. Kelley, Hyperspaces of a continuum, Trans. Amer. Math. Soc. 52 (1942), 22-36. MR 0006505 (3:315b)
  • [14] Ronald J. Knill, Cones, products, and fixed points, Fund, Math. 60 (1967), 35-46. MR 0211389 (35:2270)
  • [15] R. J. Koch and I. S. Krule, Weak cutpoint ordering on hereditarily unicoherent continua, Proc. Amer. Math. Soc. 11 (1960), 679-681. MR 0120606 (22:11356)
  • [16] J. Krasinkeiwicz, Curves which are continuous images of tree-like continua are movable, Fund. Math. 89 (1975), 233-260. MR 0388358 (52:9195)
  • [17] K. Kuratowski, Topology, Vol. II, Academic Press, New York, 1968. MR 0259835 (41:4467)
  • [18] Stefan Mazurkeiwicz, Sur l'hyperspace d'un continu, Fund. Math. 18 (1932), 171-177.
  • [19] E. E. Moise, Grille decomposition and convexification theorems for compact locally connected continua, Bull. Amer. Math. Soc. 55 (1949), 1111-1121. MR 0035430 (11:734a)
  • [20] Sam B. Nadler, Jr., A characterization of locally connected continua by hyperspace retractions, Proc. Amer. Math. Soc. 67 (1977), 167-176. MR 0458378 (56:16581)
  • [21] -, Hyperspaces of sets, Pure and Appl. Math., vol. 49, Dekker, New York, 1978.
  • [22] -, Some basic connectivity properties of Whitney map inverses in $ C(X)$, Studies in Topology, Proc. Charlotte Topology Conf., 1974 (Nick M. Stavrakas and Keith R. Allen, Eds.), Academic Press, New York, 1975, pp. 393-410. MR 0358659 (50:11118)
  • [23] -, Whitney-reversible properties, Fund. Math. 109 (1980), 235-248. MR 597070 (82b:54044)
  • [24] Ann Petrus, Contractibility of Whitney continua in $ C(X)$, General Topology and Appl. 9 (1978), 275-288. MR 510909 (80a:54010)
  • [25] H. Toruńczyk, On CE-images of the Hilbert cube and characterization of $ Q$-manifolds, Fund. Math. 106 (1980), 31-40. MR 585543 (83g:57006)
  • [26] L. E. Ward, Jr., Extending Whitney maps, preprint.
  • [27] James E. West, The subcontinua of a dendron form a Hilbert cube factor, Proc. Amer. Math. Soc. 36 (1972), 603-608. MR 0312449 (47:1006)
  • [28] Hassler Whitney, Regular families of curves. I, Proc. Nat. Acad. Sci. U.S.A. 18 (1932), 275-278.
  • [29] Gordon Thomas Whyburn, Analytic topology, Amer. Math. Soc. Colloq. Publ., vol. 28, Amer. Math. Soc., Providence, R.I., 1942. MR 0007095 (4:86b)
  • [30] M. Wojdyslawski, Sur la contractilité des hyperspaces des continus localement connexes, Fund. Math. 30 (1938), 247-252.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1982-0675074-7
Keywords: Absolute retract, acyclicity, admissible Whitney map, arc-smooth, cell-like map, contractible, dendrite, dendroid, Hilbert cube, hyperspace, locally $ n$-connected, Peano continuum, shape, smooth dendroid, starshaped, strong Whitney-reversible property, Whitney level, Whitney map, Whitney stable, $ Z$-set
Article copyright: © Copyright 1982 American Mathematical Society

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