Exercise TG.6

Question

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

Dan Whitman · Accepted Answer

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The quotient ${\mathbb{R}}/{\mathbb{Z}}$ is the reals with addition modulo one.
This is to say that two reals $x$ and $y$ are considered equivalent (i.e. they are in the same equivalence class) if $x-y \in {\mathbb{Z}}$.
This is also homeomorphic and isomorphic (with respect to the group) to the topological group $(S^1,\cdot)$ from \href{./exercise_tg-2}{Exercise~TG.2} part~(d) by the map $f: {\mathbb{R}}/{\mathbb{Z}} 	o S^1$ defined by $f(x{\mathbb{Z}}) = e^{i 2\pi x}$.
We shall not show this rigorously as doing so would be tedious, but it is not difficult to see intuitively.

Exercise TG.6

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Exercise TG.6

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