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math4301 - University of Texas at Dallas Midterm...

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University of Texas at Dallas Midterm Examination #2 Last Name: First Name and Initial: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Course Name: Mathematical Analysis 1 Number: MATH 4301 Section: 001 Instructor: Wieslaw Krawcewicz Date: November 15, 2011 Duration: 1:15 hours E-mail Address: .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Student’s Signature: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Question Weight Your Score Comments 1. 10 2. 10 3. 10 4. 10 5. 10 6. 10 7. 10 Total: 70 2
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Problem 1. Assume that ( X, d ) is a metric space. Use the appropriate logical symbols to complete the following sentences characterizing the specified topological properties. (a): the set U X is open ⇐⇒ x U ε > 0 x X d ( x, x ) < ε x U (b): the set S X is not closed ⇐⇒ x X x / S ε > 0 x S d ( x, x ) < ε (c): x A is an isolated point in A ⇐⇒ ε > 0 x A d ( x, x ) < ε x = x (d): x / A ⇐⇒ ε > 0 x A d ( x, x ) ε 3
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Problem 2. Consider a metric space ( X, d ), x o X and ε > 0. The set B ε ( x o ) := { x X : d ( x, x o ) < ε } is called an open ball at x o of radius ε . Prove that B ε ( x o ) is an open set. Hint: Clearly state the definition when a set U X is open and use it to show that B ε ( x o ) satisfies the conditions of this definition. SOLUTION: We can use the definition from Problem 1, U is open ⇐⇒ x U δ> 0 x X d ( x, x ) < ε x U. Here we have U := B ε ( x o ) and x B ε ( x o ) d ( x, x o ) < ε. We want to show that B ε ( x o ) is open. Therefore, x B ε ( x o ) δ> 0 δ := ε d ( x, x o ) > 0 (since d ( x, x o ) < ε ) such that x X d ( x, x ) < δ d ( x , x o ) d ( x , x ) + d ( x o , x ) < δ + d ( x o , x ) (by triangle inequality) = ε d ( x , x o ) + d ( x, x o ) = ε (by definition of δ ) , thus we have x B ε ( x o ) δ> 0 x X d ( x, x ) < δ x B ε ( x o ) , and consequently B ε ( x o ) is an open set.
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