# Topology Question

• May 15th 2008, 07:28 PM
Aryth
Topology Question
Let $\mathbb{R}$ and $\mathbb{R}^2$ have their respective Euclidian topologies. Endow

$S^1 = \{(x,y) \ \exists \ \mathbb{R}^2 | x^2 + y^2 = 1\}$

with subspace topology induced from $\mathbb{R}^2$. Define the relation ~ on $\mathbb{R}$ by a ~ b iff $a,b \in \mathbb{Z}$

Prove that ~ is an equivalence relation on R

Let [a] denote the ~ equivalence class containing a. Put $\frac{\mathbb{R}}{\mathbb{Z}} := \{[a] | a \in \mathbb{R}\}$ and endow $\frac{\mathbb{R}}{\mathbb{Z}}$ with the quotient topology - that is, the topology induced by the canonical projection

$n : \mathbb{R} \to \frac{\mathbb{R}}{\mathbb{Z}}$, $a \mapsto [a]$

Prove that $S^1$ is homeomorphic to $\frac{\mathbb{R}}{\mathbb{Z}}$
• May 16th 2008, 04:33 AM
KGene
The equivalent relation was given wrongly, it should be " $a\sim b$ if and only if $a-b\in \mathbb{Z}$".

Looking at $\mathbb{R}/\mathbb{Z}$, we see that there is one to one mapping between the classes in $\mathbb{R}/\mathbb{Z}$ and the interval $[0,1)$: for every class $[a]$ there is a unique real number $0\leq t<1$ such that $t\in [a]$, now map $[a]$ to the unique real number $t$ in $[0,1)$. You can show that it is a homoemorphism.

We know that $S^1$ can be parametrized by $e^{2\pi it}$ where $0\leq t<1$, now you see why $S^1\simeq [0,1)$.

Hope this helps.