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Unformatted text preview: Lecture 1 1 Review of Topology 1.1 Metric Spaces Definition 1.1. Let X be a set. Define the Cartesian product X X = { ( x, y ) : x, y X } . Definition 1.2. Let d : X X R be a mapping. The mapping d is a metric on X if the following four conditions hold for all x, y, z X : (i) d ( x, y ) = d ( y, x ), (ii) d ( x, y ) 0, (iii) d ( x, y ) = 0 x = y , and (iv) d ( x, z ) d ( x, y ) + d ( y, z ). Given a metric d on X , the pair ( X, d ) is called a metric space . Suppose d is a metric on X and that Y X . Then there is an automatic metric d Y on Y defined by restricting d to the subspace Y Y , d Y = d Y Y. (1.1)  Together with Y , the metric d Y defines the automatic metric space ( Y, d Y ). 1.2 Open and Closed Sets In this section we review some basic definitions and propositions in topology. We review open sets, closed sets, norms, continuity, and closure. Throughout this section, we let ( X, d ) be a metric space unless otherwise specified. One of the basic notions of topology is that of the open set. To define an open set, we first define the neighborhood. Definition 1.3. Given a point x o X , and a real number > 0, we define U ( x o , ) = { x X : d ( x, x o ) < } . (1.2) We call U ( x o , ) the neighborhood of x o in X . Given a subset Y X , the neighborhood of x o in Y is just U ( x o , ) Y ....
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This note was uploaded on 10/02/2010 for the course MAT unknown taught by Professor Unknown during the Fall '04 term at MIT.
 Fall '04
 unknown
 Topology

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