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Unformatted text preview: 18.100B and 18.100C Fall 2011 Final exam review sheet Most of the final will consist of problems from this list. 1. Prove that R 2 is not the union of a countable family of lines. 2. Problem 10 from page 44. 3. Let E ⊂ R be uncountable. (a) Prove that E has uncountably many limit points. (b) Prove that there exists x ∈ R such that E ∩ (∞ , x ) and E ∩ ( x, ∞ ) are both uncountable. 4. Let X be a metric space and ( K n ) n ∈ N a sequence of nonempty compact subsets of X with K n +1 ⊂ K n for all n ∈ N . Prove that if U is an open set with T ∞ n =1 K n ⊂ U , then there exists N ∈ N such that K N ⊂ U . 5. Problem 8 from page 99. 6. Let f : [0 , 1] × [0 , 1] → R be continuous, and let g : [0 , 1] → R be defined by g ( x ) = max { f ( x, y ): y ∈ [0 , 1] } . Prove that g is continuous. 7. Let K be a nonempty compact metric space with metric d , and suppose f : K → K obeys d ( f ( x ) , f ( y )) < d ( x, y ) for all x, y ∈ K with x 6 = y ....
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 Fall '10
 Prof.KatrinWehrheim
 Topology, Continuous function, Metric space, CN, Topological space, Compact space

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