# Makeup exam2 sol - University of Texas at Dallas Midterm...

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Unformatted text preview: University of Texas at Dallas Midterm Examination #2a Last Name: First Name and Initial: ...................................... ............................................ Course Name: Mathematical Analysis 1 Number: MATH 4301 Section: 001 Instructor: Wieslaw Krawcewicz Date: November 23, 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): An open ball in X (centered at x o with radius ε > 0) is the set B ε ( x o ) := { x ∈ X : d ( x o , x ) < ε } (b): the set U ⊂ X is open ⇐⇒ ∀ x ∈ U ↑ ∃ ε > B ε ( x ) ⊂ U ↑ (c): Use the definition of an open set (see 1 (b)) to prove the following state- ment: Proposition 1 Let U and V be two open sets in a metric space ( X, d ) . Then U ∩ V is also open. Proof: Assume that x ∈ U ∩ V ⇔ x ∈ U ∧ x ∈ V . Since U and V are open, { ∃ ε 1 > B ε 1 ( x ) ⊂ U ∃ ε 2 > B ε 1 ( x ) ⊂ V = ⇒ ∃ ε =min( ε 1 ,ε 2 ) > { B ε ( x ) ⊂ B ε 1 ( x ) ⊂ U B ε ( x ) ⊂ B ε 2 ( x ) ⊂ V = ⇒ B ε ( x ) ⊂ U ∧ B ε ( x ) ⊂ V = ⇒ B ε ( x ) ⊂ U ∩ V, therefore, we have ∀ x ∈ U ∩ V ∃ ε> B ε ( x ) ⊂ U ∩ V, and consequently U ∩ V is open. 3 Problem 2. Assume that ( X, d ) is a metric space. Use the appropriate logical symbols to complete the following sentences characterizing the specified topological properties.topological properties....
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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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Makeup exam2 sol - University of Texas at Dallas Midterm...

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