Normal subgroup; topological group

[tomboi03]

The integers Z are a normal subgroup of (R,+). The quotient R/Z is a familiar topological group; what is it?

okay... i attempted this problem...
and I don't know if i did it right... but can you guys check it?

Thanks~

R/Z is a familiar topological group
and Z are a normal subgroup of (R,+)
then the quotient R/Z is also a topological group with the quotient topology.
The R/Z is Hausdorff iff Z is a closed subset of (R,+) which in this case is true.
Therefore, R/Z is a quotient topology.

sorry if there are a lot of things that didn't make sense in that proof but i tried... hahaha

 

[matt grime]

That's not what it's after. It's asking you to identify it (name it, if you will) as a topological group. What topological groups do you know?

You're taking the real line and identifying x and y if x-y is what? I.e. x and y differ by what? What is a good choice for a set of equivalence classes of cosets?