Equivalence Relation and the Unit Circle: Understanding R/Z and S^1

[pivoxa15]

Homework Statement

If Z acts on R by n.x=n+x then R/Z is just S^1. CLaims the book

But I think R/Z is (0,1)

The Attempt at a Solution

Any number greater than or equal to 1 is dealt with by the equivalence relation. How does the unit circle come into it? We are dealing only with one dimensional space here.

 

[genneth]

Because the points x=0 and x'=1 are identified as the same point in R/Z?

 

[HallsofIvy]

Homework Statement

If Z acts on R by n.x=n+x then R/Z is just S^1. CLaims the book

But I think R/Z is (0,1)

The Attempt at a Solution

Any number greater than or equal to 1 is dealt with by the equivalence relation. How does the unit circle come into it? We are dealing only with one dimensional space here.

That's the whole point- the equivalence relation makes 0 and 1 equivalent- you are bending [0,1] (not (0,1)) back on itself so it becomes a circle.

 

[pivoxa15]

Actually the interval should be [0,1) so if you include 1 then 1 is 0 so it bends back on itself. However they described it as S^1. Why S^1? That is the unit circle with radius 1. So has circumference 2pi. But our interval has length 2pi?