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1998 | 156 | 3 | 261-278
Tytuł artykułu

The structure of atoms (hereditarily indecomposable continua)

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let X be an atom (= hereditarily indecomposable continuum). Define a metric ϱ on X by letting $ϱ(x,y) = W(A_{xy})$ where $A_{x,y}$ is the (unique) minimal subcontinuum of X which contains x and y and W is a Whitney map on the set of subcontinua of X with W(X) = 1. We prove that ϱ is an ultrametric and the topology of (X,ϱ) is stronger than the original topology of X. The ϱ-closed balls C(x,r) = {y ∈ X:ϱ ( x,y) ≤ r} coincide with the subcontinua of X. (C(x,r) is the unique subcontinuum of X which contains x and has Whitney value r.) It is proved that for any two (nontrivial) atoms and any Whitney maps on them, the corresponding ultrametric spaces are isometric. This implies in particular that the combinatorial structure of subcontinua is identical in all atoms. The set M(X) of all monotone upper semicontinuous decompositions of X is a lattice when ordered by refinement. It is proved that for two atoms X and Y, M(X) is lattice isomorphic to M(Y) if and only if X is homeomorphic to Y.
Rocznik
Tom
156
Numer
3
Strony
261-278
Opis fizyczny
Daty
wydano
1998
otrzymano
1997-03-17
poprawiono
1997-08-04
poprawiono
1998-02-02
Twórcy
autor
  • Department of Mathematics and Computer Science, University of Denver, Denver, Colorado 80208, U.S.A., rball@du.edu
autor
  • Department of Mathematics and Computer Science, University of Denver, Denver, Colorado 80208, U.S.A.
  • Department of Mathematics, University of Haifa, Haifa, Israel 31905
Bibliografia
  • [Ar-Pa] N. Aronszajn and P. Panitchpakdi, Extension of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6 (1956), 405-439.
  • [Bi] R. H. Bing, Higher dimensional hereditarily indecomposable continua, Trans. Amer. Math. Soc. 71 (1951), 267-273.
  • [Ho-Yo] J. G. Hocking and G. S. Young, Topology, Addison-Wesley, 1961.
  • [Kn] B. Knaster, Un continu dont tout sous-continu est indécomposable, Fund. Math. 3 (1922), 247-286.
  • [Kra] J. Krasinkiewicz, On mappings with hereditarily indecomposable fibers, Bull. Polish Acad. Sci. Math. 44 (1996), 147-156.
  • [Ku] K. Kuratowski, Topology, Volume II, Academic Press and PWN, 1968.
  • [Lev] M. Levin, Certain finite dimensional maps and their application to hyperspaces, Israel J. Math., to appear.
  • [Lev-St1] M. Levin and Y. Sternfeld, Mappings which are stable with respect to the property dimf(X) ≥ k, Topology Appl. 52 (1993), 241-265.
  • [Lev-St2] M. Levin and Y. Sternfeld, Monotone basic embeddings of hereditarily indecomposable continua, ibid. 68 (1996), 241-249.
  • [Lev-St3] M. Levin and Y. Sternfeld, Atomic maps and the Chogoshvili-Pontrjagin claim, Trans. Amer. Math. Soc., to appear.
  • [Lev-St4] M. Levin and Y. Sternfeld, The space of subcontinua of a two-dimensional continuum is infinite dimensional, Proc. Amer. Math. Soc. 125 (1997), 2771-2775.
  • [Lew] W. Lewis, The Pseudo-Arc, Marcel-Dekker, in preparation.
  • [Li] J. Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. 48 (1964).
  • [Li-Tz] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, Lecture Notes in Math. 338, Springer, 1973.
  • [Ma] S. Mazurkiewicz, Sur les continus indécomposables, Fund. Math. 10 (1927), 305-310.
  • [Na] S. B. Nadler, Jr., Hyperspaces of Sets, Marcel Dekker, 1978.
  • [Ni] J. Nikiel, Topologies on pseudo-trees and applications, Mem. Amer. Math. Soc. 416 (1989).
  • [Po1] R. Pol, A two-dimensional compactum in the product of two one-dimensional compacta which does not contain any rectangle, Topology Proc. 16 (1991), 133-315.
  • [Po2] R. Pol, On light mappings without perfect fibers on compacta, Tsukuba Math. J. 20 (1996), 11-19.
  • [Sch] W. H. Schikhof, Ultrametric Calculus. An introduction to p-adic analysis, Cambridge Univ. Press, 1984.
  • [St] Y. Sternfeld, Stability and dimension - a counterexample to a conjecture of Chogoshvili, Trans. Amer. Math. Soc. 340 (1993), 243-251.
  • [VR] A. C. M. Van Rooij, Non-Archimedean Functional Analysis, Marcel Dekker, 1978.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-fmv156i3p261bwm
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