final-answers

final-answers - 7510 Fall 06 Final Answers B1 Let L be a...

This preview shows pages 1–2. Sign up to view the full content.

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 7510 Fall 06 Final Answers B1. Let L be a straight line in the plane. The topology that L inherits as a subspace of R lscript × R lscript is a familiar topology; what is it? (The answer will be different for different lines.) Justify your answer. Solution. Consider a basis element B = [ a 1 ,b 1 ) × [ a 2 ,b 2 ) for R 2 lscript . Sup- pose that L has negative slope. Then L ∩ B = { a 1 × a 2 } if a 1 × a 2 ∈ L , so every singleton is open in L and L has the discrete topology. Oth- erwise L ∩ B is either empty or is sent to an interval [ a,b ) under some fixed (linear) bijection L → R , so L is homeomorphic to R lscript . square B2. Define an equivalence relation on R 2 by x 1 × y 1 ∼ x 2 × y 2 if x 2 1 + y 2 1 = x 2 2 + y 2 2 . The quotient space R 2 / ∼ is a familiar space; what is it? Justify your answer. Solution. Let p : R 2 → R 2 / ∼ be the quotient map. Define f : R 2 → [0 , ∞ ) by f ( x × y ) = x 2 + y 2 . Then f is a continuous surjection, and f ( x 1 × y 1 ) = f ( x 2 × y 2 ) iff x 1...
View Full Document

{[ snackBarMessage ]}

Page1 / 2

final-answers - 7510 Fall 06 Final Answers B1 Let L be a...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online