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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.
 Fall '14
 AmandaFolsom
 Geometry, Fractal Geometry, Limits, The Land

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