Lemma 232 if t r2 r2 is a d contraction with

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Unformatted text preview: ce of nonempty compact sets is a nonempty compact set. Part (4) is a generalization of a familiar fact from calculus: the intersection of a nested sequence of nonempty closed intervals is a nonempty closed interval, or a point. Note the importance of both the closed and bounded aspects of compactness. The nested sequence {(0, 1/n) : n = 1, 2, 3, . . . } of bounded sets (open as subsets of the real line) has empty intersection. Also, the nested sequence {[n, ∞) : n = 1, 2, 3, ...} of unbounded sets (closed as subsets of the real line) has empty intersection. Because T (A) = T1 (A) ∪ ... ∪ Tn (A), to show T is a contraction in the Hausdorff metric h, we need to understand how h behaves with respect to unions of compact sets. The relation was illustrated implicitly in Example 2.3.1. We make this explicit in Example 2.3.2. Example 2.3.2 The Hausdorff metric and unions Continuing with the sets and transformations of Example 2.3.1, we compare h(A1 ∪ A2 , B1 ∪ B2 ), h(A1 , B1 ), and h(A2 , B2 ). App...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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