0

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 $S^1$?

Shamisen Expert
  • 8,092
  • 1
    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 01 '16 at 06:43
  • One definition amongst others: $$S^1=\mathbb R/\mathbb Z.$$ – Did Feb 01 '16 at 07:16
  • 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 01 '16 at 09:55

2 Answers2

3

There are several definitions, which are all in some sense equivalent. People use the definition that makes thing easier to write down and that depends on what they want to do.

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 $\mathbb{R}^2$.

If you work in topology you would probably choose a definition that does not make use of an ambient space or a norm.

There are also a lot of other fields that like the complex analysis definition, as it turns the circle into an object with an algebraic structure: multiplication (it becomes a group).

All these definitions are equivalent in SOME sense. For example, they all have (canonical) topologies, which are homoeomorphic (i.e., equivalent in the sense of topology). As another example, the first two have (canonical) geometric structures, which are isomorphic (equivalent in the sense of geometry).

Bananach
  • 7,934
2

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 $\mathbb{R}^{1}\times S^{1}$.

See also the topological definition here.

Ng Chung Tak
  • 18,990