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

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