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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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