# The structure of atoms (hereditarily indecomposable continua)

Fundamenta Mathematicae (1998)

• Volume: 156, Issue: 3, page 261-278
• ISSN: 0016-2736

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## Abstract

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Let X be an atom (= hereditarily indecomposable continuum). Define a metric ϱ on X by letting $\varrho \left(x,y\right)=W\left({A}_{xy}\right)$ 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.

## How to cite

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Ball, R., Hagler, J., and Sternfeld, Yaki. "The structure of atoms (hereditarily indecomposable continua)." Fundamenta Mathematicae 156.3 (1998): 261-278. <http://eudml.org/doc/212272>.

@article{Ball1998,
abstract = {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.},
author = {Ball, R., Hagler, J., Sternfeld, Yaki},
journal = {Fundamenta Mathematicae},
keywords = {atoms (hereditarily indecomposable continua); ultrametric spaces; isometries; lattices; lattice isomorphism; atom; ultrametric; Whitney map; hereditarily indecomposable; upper semicontinuous decomposition; lattice; isometry},
language = {eng},
number = {3},
pages = {261-278},
title = {The structure of atoms (hereditarily indecomposable continua)},
url = {http://eudml.org/doc/212272},
volume = {156},
year = {1998},
}

TY - JOUR
AU - Ball, R.
AU - Hagler, J.
AU - Sternfeld, Yaki
TI - The structure of atoms (hereditarily indecomposable continua)
JO - Fundamenta Mathematicae
PY - 1998
VL - 156
IS - 3
SP - 261
EP - 278
AB - 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.
LA - eng
KW - atoms (hereditarily indecomposable continua); ultrametric spaces; isometries; lattices; lattice isomorphism; atom; ultrametric; Whitney map; hereditarily indecomposable; upper semicontinuous decomposition; lattice; isometry
UR - http://eudml.org/doc/212272
ER -

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