math4301 - University of Texas at Dallas Midterm...

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Unformatted text preview: 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: . .......................................... . Question Weight Your Score Comments 1. 10 2. 10 3. 10 4. 10 5. 10 6. 10 7. 10 Total: 70 2 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 > ∀ x ′ ∈ X ↑ d ( x, x ′ ) < ε ⇒ x ′ ∈ U (b): the set S ⊂ X is not closed ⇐⇒ ∃ x ∈ X x / ∈ S ↑ ∧ ∀ ε > ∃ x ′ ∈ S ↑ d ( x, x ′ ) < ε (c): x ∈ A is an isolated point in A ⇐⇒ ∃ ε > ∀ x ′ ∈ A ↑ d ( x, x ′ ) < ε ⇒ x ′ = x (d): x / ∈ A ⇐⇒ ∃ ε > ∀ x ′ ∈ A ↑ d ( x, x ′ ) ≥ ε ↑ 3 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> ∀ 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> δ := ε − d ( x, x o ) > (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...
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This note was uploaded on 02/23/2012 for the course MATH 4301 taught by Professor Krwaw during the Fall '11 term at King Mongkut's Institute of Technology Ladkrabang.

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math4301 - University of Texas at Dallas Midterm...

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