# What is the official definition of $S^1$ [closed]

I need an official definition of $S^1$ that is better than $\{circle\}$.

The reason is because I am interesting in defining a function $f: \mathbb{R} \to S^1$ where $\mathbb{R}$ is the interval $[0, 2\pi)$, but I do not know what the image is

I tried to come up with one:

$S^1 = \{x|\|x\| = 1\}$

But there seems to be better ones out there.

Can someone please provide with an official def of $S^1$?

## closed as unclear what you're asking by Did, user91500, Fabian, Claude Leibovici, Chris GodsilFeb 1 '16 at 12:48

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• I find you choice of phrase a bit amusing. What could be the authority that makes a definition official?? Anyway, what I find lacking from your suggestion is the ambient space. $S^1$ consists of the points $x$ with norm $=1$. Points in ... what space? Some seventeen dimensional monster? To get $S^1$ you need to specify the space as $\Bbb{R}^2$ and the norm as the usual Euclidean norm (rather than, say, the taxicab norm). – Jyrki Lahtonen Feb 1 '16 at 6:43
• One definition amongst others: $$S^1=\mathbb R/\mathbb Z.$$ – Did Feb 1 '16 at 7:16
• There is no "official" definition, there are several of them, all equivalent, among which is to be found the one that you yourself have given, which is absolutely correct. You may also try $S^1 = \{ z \in \Bbb C \mid |z| = 1 \}$, very similar to yours but with $\Bbb R ^2$ viewed as $\Bbb C$ and thus with $\| \cdot \|$ replaced by the modulus. – Alex M. Feb 1 '16 at 9:55

In general, $S^{n}=\{ \boldsymbol{x} \in \mathbb{R}^{n+1}: |\boldsymbol{x}| =1 \}$.

$S^{0}$ is two isolated points, $S^{1}$ is a circle, $S^{2}$ is a sphere and $S^{3}$ is a 3-sphere.

Moreover, a torus $T^{2}$ is $S^{1}\times S^{1}$ and (infinite) cylinder is $\mathbb{R}^{1}\times S^{1}$.

If you work in complex analysis, you might find it nice to set $S^1=\{e^{i2\pi x} : x\in [0,1)\}\subset\mathbb{C}$.
If you do geometry you might go with the definition you presented, with ambient space $\mathbb{R}^2$.