I encountered the quotient $\mathbb{T} = \mathbb{R}/2\pi\mathbb{Z}$ while beginning to study harmonic analysis. Despite having taken elementary abstract algebra and number theory, I am still having difficulty really seeing what is occurring in the construction of a quotient.
I decided to construct the simplest quotient I knew of $\mathbb{Z}/2\mathbb{Z}$. Please correct me if I make any mistakes: So we start with the additive group of integers, $\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}$. Next we have a normal subgroup, the even integers $2\mathbb{Z} = \{\ldots, -4, -2, 0, 2, 4, \ldots\}$. Now we have an obvious homomorphism between the two, $\phi(n) = 2n$.
Now here is where I am having difficulty - the quotient $\mathbb{Z}/2\mathbb{Z}$ is a group of cosets of the normal subgroup $2\mathbb{Z}$, $\{\overline{0}, \overline{1}\}$ and it partitions the integers.
I understand how $2\mathbb{Z}$ is a normal subgroup of $\mathbb{Z}$ and the homomorphism between the two, but I can't wrap my mind around how we then go to a group of just two cosets of $2\mathbb{Z}$.
Could someone explain this, as well as how we use this same idea with the quotient $\mathbb{T}$?