p1.pdf - Math 25b Honors Linear Algebra and Real Analysis II Homework Assignment#1(31 January 2014 Metric topology basics “I’m sorry ” “Don’t

p1.pdf - Math 25b Honors Linear Algebra and Real Analysis...

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Math 25b: Honors Linear Algebra and Real Analysis II Homework Assignment #1 (31 January 2014): Metric topology basics “I’m sorry. . . ” “Don’t topologize.” —Martin Gardner, The Unexpected Hanging Definition and constructions of metric spaces: 1. For any set X define the discrete metric on X by d ( p, q ) = 0 if p = q and d ( p, q ) = 1 if p 6 = q . (See Simmons, page 51, Example 1.) Prove that this is indeed a metric. With this metric, which subsets of X are open? Which are closed? Which are dense? 2. [Simmons, page 58, Problem 1] Let ( X, d ) be a metric space. Define d 0 ( x, y ) := d ( x, y ) 1 + d ( x, y ) for all x, y X . i) Prove that d 0 is also a metric on X . ii) Prove that a subset of X is open under the metric d if and only if it is open under d 0 . [Thus ( X, d ) and ( X, d 0 ) are the same as “topological spaces”, but generally not isometric (identical as metric spaces).] iii) Show that the metric space ( X, d 0 ) is always bounded, even though ( X, d ) might not be. 3. Which of the following defines a metric on R ? Explain. (For a hint see Problem 10.) i) d 1 ( x, y ) := ( x - y ) 2 ii) d 2 ( x, y ) := | x 2 -
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