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Unformatted text preview: Lecture 6 6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a nonempty set X is a map d : X X [0 , ) with the following properties: (i) If x,y X are points with d ( x,y ) = 0, then x = y ; (ii) d ( x,y ) = d ( y,x ), for all x,y X ; (iii) d ( x,y ) d ( x,z ) + d ( y,z ), for all x,y,z X . A metric space is a pair ( X,d ), where X is a set, and d is a metric on X . Notations. If ( X,d ) is a metric space, then for any point x X and any r > 0, we define the open and closed balls: B r ( x ) = { y X : d ( x,y ) < r } , B r ( x ) = { y X : d ( x,y ) r } . Definition. Suppose ( X,d ) is a metric space. Then X carries a natural toplogy constructed as follows. We say that a set D X is open , if it has the property: for every x D , there exists some r x > , such that B r x ( x ) D . One can prove that the collection T d = { D X : D open } is indeed a topology , i.e. we have and X are open; if ( D i ) i I is a family of open sets, then S i I D i is again open; if D 1 and D 2 are open, then D 1 D 2 is again open. The topology thus constructed is called the metric topology . Remark 6.1. Let ( X,d ) be a metric space. Then for every p X , and for every r > 0, the set B r ( p ) is open, and the set B r ( p ) is closed. If we start with some x B r ( p ), an if we define r x = r d ( x,p ), then for every y B r x ( x ) we will have d ( y,p ) d ( y,x ) + d ( x,p ) < r x + d ( x,p ) = r, so y belongs to B r ( p ). This means that B r x ( x ) B r ( p ). Since this is true for all x B r ( p ), it follows that B r ( p ) is indeed open. 31 32 LECTURE 6 To prove that B r ( p ) is closed, we need to show that its complement X r B r ( p ) = { x X : d ( x,p ) > r } is open. If we start with some x X r B r ( p ), an if we define x = d ( p,x ) r , then for every y B x ( x ) we will have d ( y,p ) d ( p,x ) d ( y,x ) > d ( p,x ) x = r, so y belongs to X r B r ( p ). This means that B x ( x ) X r B r ( p ). Since this is true for all x X r B r ( p ), it follows that X r B r ( p ) is indeed open. Remark 6.2. The metric toplogy on a metric space ( X,d ) is Hausdorff. Indeed, if we start with two points x,y X , with x 6 = y , then if we choose r to be a real number, with < r < d ( x,y ) 2 , then we have B r ( x ) B r ( y ) = . (Otherwise, if we have a point z B r ( x ) B r ( y ), we would have 2 r < d ( x,y ) d ( x,z ) + d ( y,z ) < 2 r , which is impossible.) Remark 6.3. Let ( X,d ) be a metric space, and let M be a subset of X . Then d M M is a metric on M , and the metric topology on M defined by this metric is precisely the induced toplogy from X . This means that a set A M is open in M if and only if there exists some open set D X with A = M D ....
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This note was uploaded on 10/11/2010 for the course MATH 11 taught by Professor Smith during the Three '10 term at ADFA.
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