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1996 | 69 | 2 | 289-296
Tytuł artykułu

On uncountable collections of continua and their span

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We prove that if the Euclidean plane $ℝ^2$ contains an uncountable collection of pairwise disjoint copies of a tree-like continuum X, then the symmetric span of X is zero, sX = 0. We also construct a modification of the Oversteegen-Tymchatyn example: for each ε > 0 there exists a tree $X ⊂ ℝ^2$ such that σX < ε but X cannot be covered by any 1-chain. These are partial solutions of some well-known problems in continua theory.
Rocznik
Tom
69
Numer
2
Strony
289-296
Opis fizyczny
Daty
wydano
1996
otrzymano
1993-10-06
poprawiono
1995-02-21
Twórcy
  • University of Trieste, 1, Piazzale Europa, 34100 Trieste, Italy
  • Faculty of Mechanics And Mathematics, Moscow State University, Vorobyovy Hills, 117899 Moscow, Russia
  • Steklov Mathematical Institute, Russian Academy of Sciences, 42, Vavilova Street, 117966 Moscow, Russia
Bibliografia
  • [1] R. D. Anderson, Continuous collections of continuous curves, Duke Math. J. 21 (1954), 363-367.
  • [2] V. I. Arnold, Ordinary Differential Equations, Nauka, Moscow, 1971 (in Russian).
  • [3] B. J. Baker and M. Laidacker, Embedding uncountably many mutually exclusive continua into Euclidean space, Canad. Math. Bull. 32 (1989), 207-214.
  • [4] C. E. Burgess, Collections and sequences of continua in the plane I, II, Pacific J. Math. 5 (1955), 325-333; 11 (1961), 447-454.
  • [5] C. E. Burgess, Continua which have width zero, Proc. Amer. Math. Soc. 13 (1962), 477-481.
  • [6] P. E. Conner and E. E. Floyd, Fixed points free involutions and equivariant maps, Bull. Amer. Math. Soc. 66 (1960), 416-441.
  • [7] H. Cook, W. T. Ingram and A. Lelek, Eleven annotated problems about continua, in: Open Problems in Topology, J. van Mill and G. M. Reed (eds.), North-Holland, Amsterdam, 1990, 295-302.
  • [8] J. F. Davis, The equivalence of zero span and zero semispan, Proc. Amer. Math. Soc. 90 (1984), 133-138.
  • [9] D E. K. van Douwen, Uncountably many pairwise disjoint copies of one metrizable compactum in another, Topology Appl. 51 (1993), 87-91.
  • [10] W. T. Ingram, An uncountable collection of mutually exclusive planar atriodic tree-like continua with positive span, Fund. Math. 85 (1974), 73-78.
  • [11] H. Kato, A. Koyama and E. D. Tymchatyn, Mappings with zero surjective span, Houston J. Math. 17 (1991), 325-333.
  • [12] A. Lelek, Disjoint mappings and the span of the spaces, Fund. Math. 55 (1964), 199-214.
  • [13] P. Minc, On simplicial maps and chainable continua, Topology Appl. 57 (1994), 1-21.
  • [14] R. L. Moore, Concerning triods in the plane and the junction points of plane continua, Proc. Nat. Acad. Sci. U.S.A. 14 (1928), 85-88.
  • [15] L. G. Oversteegen, On span and chainability of continua, Houston J. Math. 15 (1989), 573-593.
  • [16] L. Oversteegen and E. D. Tymchatyn, Plane strips and the span of continua I, II, ibid. 8 (1982), 129-142; 10 (1984), 255-266.
  • [17] C. R. Pittman, An elementary proof of the triod theorem, Proc. Amer. Math. Soc. 25 (1970), 919.
  • [18] D. Repovš and E. V. Ščepin, On the symmetric span of continua, Abstracts Amer. Math. Soc. 14 (1993), 319, No. 93T-54-42.
  • [19] D. Repovš, A. B. Skopenkov and E. V. Ščepin, On embeddability of X×I into Euclidean space, Houston J. Math. 21 (1995), 199-204.
  • [20] J. H. Roberts, Concerning atriodic continua, Monatsh. Math. 37 (1930), 223-230.
  • [21] K. Sieklucki, A generalization of a theorem of K. Borsuk concerning the dimension of ANR-sets, Bull. Acad. Polon. Sci. 10 (1962), 433-463; Erratum, 12 (1964), 695.
  • [22] G. S. Young, Jr., A generalization of Moore's theorem on simple triods, Bull. Amer. Math. Soc. 5 (1944), 714.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-cmv69i2p289bwm
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