Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On a characterization of $ W$-sets and the dimension of hyperspaces


Authors: J. Grispolakis and E. D. Tymchatyn
Journal: Proc. Amer. Math. Soc. 100 (1987), 557-563
MSC: Primary 54F20; Secondary 54B20, 54F45
DOI: https://doi.org/10.1090/S0002-9939-1987-0891163-6
MathSciNet review: 891163
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A subcontinuum $ A$ of a continuum $ X$ is a $ W$-set if for each mapping $ f:Y \twoheadrightarrow X$ of an arbitrary continuum $ Y$ onto $ X$ there is a continuum in $ Y$ which is mapped by $ f$ onto $ A$. We characterize $ W$-sets in terms of accessibility by small continua. We localize several known results on continua all of whose subcontinua are $ W$-sets. Finally, we extend a result of J. T. Rogers by proving that if $ X$ is an atriodic continuum whose first Čech cohomology group is finitely generated then the hyperspace $ C(X)$ of subcontinua of $ X$ is two dimensional.


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

  • [1] J. F. Davis, Atriodic acyclic continua and Class W, Proc. Amer. Math. Soc. 90 (1984), 477-482. MR 728372 (85f:54067)
  • [2] S. Eilenberg, Sur les transformation d'espaces metriques en circonference, Fund. Math. 24 (1935), 160-176.
  • [3] J. Grispolakis and E. D. Tymchatyn, Continua which are images of weakly confluent mappings only. I, Houston J. Math. 5 (1979), 483-502. MR 567908 (81d:54008)
  • [4] -, Continua which are images of weakly confluent mappings only. II, Houston J. Math. 6 (1980), 375-387. MR 597777 (82h:54052)
  • [5] -, Weakly confluent mappings and the covering property of hyperspaces, Proc. Amer. Math. Soc. 74 (1979), 177-182. MR 521894 (80d:54041)
  • [6] -, On the Čech cohomology of continua with no $ n$-ods, Houston J. Math. 11 (1985), 505-513. MR 837989 (88c:54019)
  • [7] W. Hurewicz and H. Wallman, Dimension theory, Princeton Univ. Press, Princeton, N.J., 1948. MR 0006493 (3:312b)
  • [8] W. T. Ingram, $ C$-sets and mappings of continua, Topology Proc. 7 (1982), 83-90. MR 696623 (85i:54040)
  • [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 certain plane continua, Bull. Polish Acad. Sci. Math. 23 (1975), 981-983. MR 0493999 (58:12942)
  • [11] S. B. Nadler, Jr., Hyperspaces of sets, Dekker, New York, 1978. MR 0500811 (58:18330)
  • [12] V. C. Nall, Weak confluence and $ W$-sets, Topology Proc. 8 (1983), 161-194. MR 738474 (85d:54042)
  • [13] L. G. Oversteegen and E. D. Tymchatyn, On atriodic tree-like continua, Proc. Amer. Math. Soc. 83 (1981), 201-204. MR 620013 (82h:54058)
  • [14] J. H. Roberts and N. E. Steenrod, Monotone transformations of $ 2$-dimensional manifolds, Ann. of Math. (2) 39 (1939), 851-862.
  • [15] J. T. Rogers, Jr., Weakly confluent maps and finitely-generated cohomology, Proc. Amer. Math. Soc. 78 (1980), 436-438. MR 553390 (80m:54048)
  • [16] R. H. Sorgenfrey, Concerning triodic continua, Amer. J. Math. 66 (1944), 439-460. MR 0010968 (6:96d)
  • [17] E. D. Tymchatyn and B. O. Friberg, A problem of Martin concerning strongly convex metrics on $ {E^3}$, Proc. Amer. Math. Soc. 43 (1974), 461-464. MR 0336735 (49:1508)
  • [18] L. Vietoris, Über den höheren Zusammenhang kompakten Räume und eine Klasse van zusammenhangstreuen Abbildungen, Math. Ann. 97 (1972), 454-472. MR 1512371

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54F20, 54B20, 54F45

Retrieve articles in all journals with MSC: 54F20, 54B20, 54F45


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1987-0891163-6
Keywords: Continua, weakly confluent mappings, dimension of hyperspaces, atriodic
Article copyright: © Copyright 1987 American Mathematical Society

American Mathematical Society