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Metric spaces admitting only trivial weak contractions

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Balka R.

Fundamenta Mathematicae

2013

tom 221

nr 1

83-94

If (X,d) is a metric space then a map f: X → X is defined to be a weak contraction if d(f(x),f(y)) < d(x,y) for all x,y ∈ X, x ≠ y. We determine the simplest non-closed sets X ⊆ ℝⁿ in the sense of descriptive set-theoretic complexity such that every weak contraction f: X → X is constant. In order to do so, we prove that there exists a non-closed $F_{σ}$ set F ⊆ ℝ such that every weak contraction f: F → F is constant. Similarly, there exists a non-closed $G_{δ}$ set G ⊆ ℝ such that every weak contraction f: G → G is constant. These answer questions of M. Elekes. We use measure-theoretic methods, first of all the concept of generalized Hausdorff measure.

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1 Fundamenta Mathematicae

1 Balka R.

1 2013

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Metric spaces admitting only trivial weak contractions