Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Whitney continua of curves


Author: Hisao Kato
Journal: Trans. Amer. Math. Soc. 300 (1987), 367-381
MSC: Primary 54B20; Secondary 54F43
DOI: https://doi.org/10.1090/S0002-9947-1987-0871681-1
MathSciNet review: 871681
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we prove several theorems relating shape properties of Whitney continua of curves. In particular, we investigate the fundamental dimension and the shape type of Whitney continua of curves.


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

  • [1] K. Borsuk, Theory of shape, Monograf. Mat. No. 59, PWN, Warszawa, 1975. MR 0293602 (45:2679)
  • [2] J. H. Case and R. E. Chamberlin, Characterization of tree-like continua, Pacific J. Math. 10 (1960), 73-84. MR 0111000 (22:1868)
  • [3] R. Duda, On the hyperspace of subcontinua of a finite graph. I, II, Fund. Math. 62 (1968), 265-286; 63 (1968), 225-255. MR 0236881 (38:5175a)
  • [4] -, Correction to the paper On the hypersurface of subcontinua of a finite graph. I, Fund. Math. 69 (1970), 207-211. MR 0273575 (42:8453)
  • [5] D. Curtis, Stable points in hyperspaces of Peano continua, preprint. MR 876898 (88e:54011)
  • [6] J. T. Goodykoontz, Jr. and S. B. Nadler, Jr., Whitney levels in hypersurfaces of certain Peano continua, Trans. Amer. Math. Soc. 274 (1982), 671-694. MR 675074 (84h:54010)
  • [7] H. Kato, Concerning hyperspaces of certain Peano continua and strong regularity of Whitney maps, Pacific J. Math. 119 (1985), 159-167. MR 797021 (86j:54019)
  • [8] -, Shape properties of Whitney maps for hyperspaces, Trans. Amer. Math. Soc. 297 (1986), 529-546. MR 854083 (87h:54022)
  • [9] J. L. Kelley, Hyperspaces of a continuum, Trans. Amer. Math. Soc. 52 (1942), 22-36. MR 0006505 (3:315b)
  • [10] J. Krasinkiewicz, On the hyperspaces of snake-like and circle-like continua, Fund. Math. 83 (1974), 155-164. MR 0418058 (54:6102)
  • [11] -, Shape properties of hypersurfaces, Fund. Math. 101 (1978), 79-91. MR 512244 (80b:54038)
  • [12] J. Krasinkiewicz and S. B. Nadler, Jr., Whitney properties, Fund. Math. 98 (1978), 165-180. MR 0467691 (57:7546)
  • [13] S. Mardešić and J. Segal, Shape theory, North-Holland Mathematical Library, 1982. MR 676973 (84b:55020)
  • [14] M. Lynch, Whitney levels in $ {C_p}(X)$ are absolute retracts, preprint.
  • [15] S. B. Nadler, Jr., Hyperspaces of sets, Pure and Appl. Math., vol. 49, Dekker, New York, 1978. MR 0500811 (58:18330)
  • [16] -, Whitney-reversible properties, Fund. Math. 109 (1980), 235-248. MR 597070 (82b:54044)
  • [17] A. Petrus, Contractibility of Whitney continua in $ C(X)$, General Topology Appl. 9 (1978), 275-288. MR 510909 (80a:54010)
  • [18] J. T. Rogers, The cone = hypersurface property, Canad. J. Math. 24 (1972), 279-285. MR 0295302 (45:4370)
  • [19] -, Continua with cones homeomorphic to hyperspaces, General Topology Appl. 3 (1973), 283-289. MR 0362257 (50:14699)
  • [20] -, Dimension and Whitney subcontinua of $ C(X)$, General Topology Appl. 6 (1976), 91-100. MR 0420536 (54:8550)
  • [21] -, Embedding the hypersurfaces of circle-like plane continua, Proc. Amer. Math. Soc. 29 (1971), 165-168. MR 0273578 (42:8456)
  • [22] -, Hyperspaces of arc-like and circle-like continua, Lecture Notes in Math., vol. 375, Springer-Verlag, Berlin and New York, 1974, pp. 231-234. MR 0356001 (50:8474)
  • [23] -, Whitney continua in the hyperspaces $ C(X)$, Pacific J. Math. 58 (1975), 569-584.
  • [24] -, Applications of a Vietoris-Begle theorem for multi-valued maps to the cohomology of hyperspaces, Michigan Math. J. 22 (1975), 315-319. MR 0397678 (53:1536)
  • [25] J. Segal, Hyperspaces of the inverse limit space, Proc. Amer. Math. Soc. 10 (1959), 706-709. MR 0108780 (21:7492)
  • [26] L. E. Ward, Jr., Extending Whitney maps, Pacific J. Math. 93 (1981), 465-469. MR 623577 (82m:54003)
  • [27] H. Whitney, Regular families of curves. I, Proc. Nat. Acad. Sci. U.S.A. 18 (1939), 184-192.
  • [28] S. Nowak, Some properties of fundamental dimension, Fund. Math. 85 (1974), 211-227. MR 0383344 (52:4225)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 54B20, 54F43

Retrieve articles in all journals with MSC: 54B20, 54F43


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1987-0871681-1
Keywords: Curve, inverse limit, hyperspace, Whitney map, Whitney continua, FANR, movable, (strongly) winding curve, $ \theta (m)$-curve, tree-like, circle-like
Article copyright: © Copyright 1987 American Mathematical Society

American Mathematical Society