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

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

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.

Page1 / 11

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

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online