The structure of atoms (hereditarily indecomposable continua)

• Autorzy: R. N. Ball , J. N. Hagler , Yaki Sternfeld
• 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.

Słowa kluczowe

EN

atoms (hereditarily indecomposable continua); ultrametric spaces; isometries; lattices; lattice isomorphism

Kategorie tematyczne

• 54C35: Function spaces
• 54F15: Continua and generalizations
• 54B20: Hyperspaces
• 54F45: Dimension theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1998
• Tom: 156
• Numer: 3
• Strony: 261-278

Daty

wydano

1998

otrzymano

1997-03-17

poprawiono

1997-08-04

poprawiono

1998-02-02

Twórcy

autor

R. N. Ball

• Department of Mathematics and Computer Science, University of Denver, Denver, Colorado 80208, U.S.A.,

autor

J. N. Hagler

• Department of Mathematics and Computer Science, University of Denver, Denver, Colorado 80208, U.S.A.

autor

Yaki Sternfeld

• 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.