The structure of atoms (hereditarily indecomposable continua)

Ball, R.; Hagler, J.; Sternfeld, Yaki

Fundamenta Mathematicae, Tome 158 (1998), p. 261-278 / Harvested from The Polish Digital Mathematics Library

Résumé

Let X be an atom (= hereditarily indecomposable continuum). Define a metric ϱ on X by letting $ϱ (x, y) = W (A_{x y})$ 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.

Publié le : 1998-01-01

Zbl 0906.54026

EUDML-ID : urn:eudml:doc:212272

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     author = {R. Ball and J. Hagler and Yaki Sternfeld},
     title = {The structure of atoms (hereditarily indecomposable continua)},
     journal = {Fundamenta Mathematicae},
     volume = {158},
     year = {1998},
     pages = {261-278},
     zbl = {0906.54026},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv156i3p261bwm}
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Ball, R.; Hagler, J.; Sternfeld, Yaki. The structure of atoms (hereditarily indecomposable continua). Fundamenta Mathematicae, Tome 158 (1998) pp. 261-278. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv156i3p261bwm/

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