Circle as a quotient of the real line

Example

On $\mathbb{R}$, let $s \sim t$ when $s - t \in \mathbb{Z}$. Let $\mathbb{R} / \mathbb{Z}$ be the set of classes, with the quotient topology. Let $S^1 = \{ (x, y) \in \mathbb{R}^2 : x^2 + y^2 = 1 \}$ be the unit circle, with the subspace topology of $\mathbb{R}^2$. Then the map $[t] \mapsto (\cos 2\pi t, \sin 2\pi t)$ is a homeomorphism from $\mathbb{R} / \mathbb{Z}$ to $S^1$.

Remarks

The map is well defined and continuous by the universal property of the quotient topology, and it is bijective. The space $\mathbb{R} / \mathbb{Z}$ is compact, because it is the image of $[0, 1]$ under the quotient map. The circle is Hausdorff. Thus the map is a homeomorphism, by this theorem. Equivalently, $S^1$ is homeomorphic to $[0, 1]$ with the two endpoints $0$ and $1$ glued together. Addition of classes makes $\mathbb{R} / \mathbb{Z}$ a group, the circle group.

Explanations

L1 human

Wrap the real line around a circle of length $1$. Points that differ by an integer land at the same place. The quotient $\mathbb{R} / \mathbb{Z}$ records exactly this wrapping.