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

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SUMMARY

The discussion centers on the equivalence relation defined by Z acting on R, specifically that R/Z is equivalent to S^1, the unit circle. Participants clarify that the equivalence relation identifies the endpoints 0 and 1 in the interval [0,1), effectively bending it into a circular shape. The confusion arises from the relationship between the interval's length and the circumference of the unit circle, which is 2π. Ultimately, the conclusion is that R/Z indeed corresponds to S^1 due to this identification.

PREREQUISITES
  • Understanding of equivalence relations in mathematics
  • Familiarity with the concepts of R/Z and S^1
  • Basic knowledge of topology and one-dimensional spaces
  • Comprehension of interval notation, specifically [0,1) versus (0,1)
NEXT STEPS
  • Study the properties of equivalence relations in abstract algebra
  • Explore the topology of the unit circle S^1 and its applications
  • Learn about the concept of quotient spaces in topology
  • Investigate the relationship between intervals and their geometric representations
USEFUL FOR

Mathematics students, particularly those studying topology and abstract algebra, as well as educators seeking to clarify the relationship between equivalence relations and geometric representations.

pivoxa15
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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.
 
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Because the points x=0 and x'=1 are identified as the same point in R/Z?
 
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.
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.
 
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?
 

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