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The structure of atoms (hereditarily indecomposable continua)

R. BallJ. HaglerYaki Sternfeld — 1998

Fundamenta Mathematicae

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...

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