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Unformatted text preview: ment holds for every point p ∈ (Aδ1 )δ2 , so (Aδ1 )δ2 ⊆ Aδ1 +δ2 . Proof of Prop. 2.2.1. We must show h satisfies positive-definiteness, symmetry, and the triangle inequality. Positive-definiteness Because ǫ-thickenings are defined only for ǫ ≥ 0, for all compact sets A and B , we see h(A, B ) ≥ 0. To show h(A, B ) = 0 if and only if A = B , first suppose h(A, B ) = 0. Then A ⊆ Bǫ and B ⊆ Aǫ for all ǫ > 0. It follows that A = B , because if not, then there is some a ∈ A − B , say. (Similar argument if it’s b ∈ B − A.) Compactness guarantees there is some ǫ > 0 for which d(a, b) ≥ ǫ for all b ∈ B , contradicting h(A, B ) < ǫ. To show A = B implies h(A, B ) = 0, suppose h(A, B ) = ǫ > 0. Then A ⊆ Bǫ and A Bǫ′ for all ǫ′ < ǫ. (Or the roles of A and B may be reversed. The argument is clear.) Because A and B are compact, there is a point a ∈ A with d(a, b) ≥ ǫ for all b ∈ B , and so A = B . Symmetry See Prob. 2.2.3. Triangle inequality Given any three comp...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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