1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Quotient topology

  1. Apr 4, 2010 #1
    1. The problem statement, all variables and given/known data

    verify that R, the reals, quotiented by the equivalence relation x~x+1 is S^1

    2. Relevant equations

    3. The attempt at a solution

    All i can think of is to draw a unit square and identify sides like the torus, but this would be using IxI, a subset of R^2, and gives a cylinder at best... Obviously I am missing the point, so any help would be great.
  2. jcsd
  3. Apr 4, 2010 #2
    Ok thought about this some more.

    What I have done above is actually on the right path, if the horizontals of the square are identified we get a cylinder, if the vertical height is shrunk to nothing, so we are working in just R, then we get a circle. Then the result follows. I think the logic I present is correct but is there a more mathematically concise way to present this?
  4. Apr 5, 2010 #3
  5. Apr 5, 2010 #4


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Sounds like you should try to write down a function from from R/~ into S1 (or from S1 into R/~) and then try to prove that it's a homeomorphism. You mentioned the "quotient topology" in the title. I take it that means that the open sets on R/~ are defined to be the preimages [itex]\pi^{-1}(U)[/itex] of open sets U in R, where [itex]\pi:\mathbb R\rightarrow\mathbb R/\sim[/itex] is the function that takes a number to its equivalence class: [itex]\pi(x)=[x][/itex].
  6. Apr 7, 2010 #5
    Ok so the quotient R/~ = [0,1) for the relation x~x+1?

    Then defining the map f:[0,1)---S^1 via


    for x in [0,1).

    Yes I am working on the definition that open sets in the preimage are open defines continuity and so give definition of homeomorphism.

    I take by showing that S^1 is homeomorphic to unit interval this shows that it has the same topology as the real line?
  7. Apr 7, 2010 #6


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    What I'm suggesting is that you define a topology on R/~ by saying that the function [itex]\pi[/itex] is continuous. Then you you show that your f is continuous with respect to that that topology on R/~.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook