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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