The discussion revolves around the concept of equivalence relations in the context of the real numbers and the unit circle, specifically examining the relationship between R/Z and S^1. Participants are exploring how the equivalence relation defined by Z acting on R leads to the identification of points and the implications for the structure of R/Z.
The discussion is ongoing, with participants actively questioning the definitions and implications of the equivalence relation. There is a recognition of the need to clarify why R/Z is described as S^1, and various interpretations of the intervals are being explored without a clear consensus.
Participants note that the interval should be considered as [0,1) and discuss the implications of including or excluding endpoints in the context of the equivalence relation. There is also mention of the length of the interval in relation to the circumference of the unit circle.
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 said: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.