Turn on thread page Beta
    • Thread Starter
    Offline

    2
    ReputationRep:
    How would I go about proving that an annulus in R^2 is path connected? It seems like a mess. I tried to make 4 central points (a+b/2,0), (0, a+b/2), (-(a+b/2),0), (0, -(a+b/2)) (where the annulus is {(x,y) : a^2 < x^2 + y^2 < b^2} and show that these 4 points are path connected and then "obviously" any point in the annulus can be path connected to one of these central points. But this seemed too fiddly and difficult.
    Offline

    13
    ReputationRep:
    (Original post by gangsta316)
    How would I go about proving that an annulus in R^2 is path connected? It seems like a mess. I tried to make 4 central points (a+b/2,0), (0, a+b/2), (-(a+b/2),0), (0, -(a+b/2)) (where the annulus is {(x,y) : a^2 < x^2 + y^2 < b^2} and show that these 4 points are path connected and then "obviously" any point in the annulus can be path connected to one of these central points. But this seemed too fiddly and difficult.
    It's not obvious that any point can be path connected to one of those four. (well... it is... but the proof seems about as hard as the general case).

    I'd do it directly. If you think of R^2 as the complex numbers (just so I can use the concept of an argument), have the argument and modulus vary as some parameter t goes from 0 to 1. So if P1 and P2 are points in the annulus, when t = 0, you're at point P1, and when t=1 you're at point P2.
 
 
 
Poll
Is the Big Bang theory correct?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.