Verifying S1 in Quotient Topology of R with x~x+1

In summary, the conversation discusses verifying that the real numbers, quotiented by the equivalence relation x~x+1, is homeomorphic to the unit circle, S^1. The suggested approach is to define a topology on R/~ by stating that the function \pi is continuous, and then showing that the function f defined as f(x)=exp(2*Pi*x*i) is also continuous with respect to this topology. This will demonstrate that S^1 has the same topology as the real line.
  • #1

Homework Statement

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

Homework Equations

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.
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  • #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
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].
  • #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?
  • #6
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/~.

Suggested for: Verifying S1 in Quotient Topology of R with x~x+1