How to show that ${\mathbb R}/{\mathbb Z}$ is a compact topological space? I'm trying to understand the quotient space ${\mathbb R}/{\mathbb Z}$ (with the quotient topology) and I am stuck with the following question:

How can I show that ${\mathbb R}/{\mathbb Z}$ is compact?

One can either establish a homeomorphism between ${\mathbb R}/{\mathbb Z}$ and the set $\{(x,y):x^2+y^2=1\}$ or directly show by definition of compactness. In either way I don't know how to go on. Or is there a handy theorem that one can use here?

[Added:] Thanks to David's comment and Daniel's elaboration, one should note that in this post $\mathbb{Z}$ should be understood as a group acting on $\mathbb{R}$. More explicitly, 

Consider the set $X=\mathbb {R}$  of all real numbers with the ordinary topology, and write $x \sim y$ if and only if $x − y$ is an integer. Then the quotient space $X/\sim$ is homeomorphic to the unit circle $S^1$ via the homeomorphism which sends the equivalence class of $x$ to $\exp(2\pi ix)$. More details are in the Wikipedia article. 

 A: $\mathbb{R}/\mathbb{Z}$ has a (minor) inconvenience, which is that it doesn't come from a compact space, so we can't guarantee compactness straightforwardly, and neither can we use the quotient map directly to exhibit a homeomorphism (although this is easily manageable in this special case), since we only have a continuous bijection a priori. But in order to avoid that inconvenience, one can do the following:
Prove that $[0,1]/\sim $, where $0 \sim 1$, is homeomorphic to $S^1$. For that, consider $f:[0,1] \to S^1 $ given by $f(t)=e^{2\pi i t}$, then pass to the quotient using the universal property. It is easily verified to be a bijection in the quotient. Having compact domain (since $[0,1]/ \sim=\pi([0,1])$ and $\pi$ is continuous), it is a homeomorphism (since $S^1$ is Hausdorff). 
Done that, now prove that $[0,1]/ \sim$ is homeomorphic to $\mathbb{R}/\mathbb{Z}$. For that, consider $f:[0,1]\to \mathbb{R}$ the inclusion, compose with the quotient to yield a function $[0,1]\to \mathbb{R}/\mathbb{Z}$, then use the universal property to get a function $[0,1]/\sim \to \mathbb{R}/\mathbb{Z}$, which is a continuous bijection from a compact set to a Hausdorff set again.
A: The map
$$
f\colon [0,1]\to \mathbb{R}/\mathbb{Z},
\qquad
f(x)=x+\mathbb{Z}
$$
is continuous and surjective.
