Am I understanding the unit Torus? As as I read in some discrete mathematics papers, the unit torus in $\mathbb{R^n}$ is defined as:
$$T^n = \mathbb{R^n}/\mathbb{Z^n}$$
So are the classes in $\mathbb{R^n}$ modulo $\mathbb{Z^n}$.
For $n = 1$ and for $n = 2$ this would mean, respectively:
$$T = \{ x \in (-1, 1) \} \quad T^2 = \{ (x, y) \in \mathbb{R^2} : x^2 + y^2 < 1 \}$$ 
This is because an element $n \in \mathbb{Z}$ would be in the class of $0$, and an element $w \in \mathbb{R}-\mathbb{Z}$ can be written as $w = x + n$ for $x \in (-1, 1) $ and $n \in \mathbb{N}$, so $w$ would be in some class of $(-1, 1)$ (So there are infinite classes).
Am I understanding this concepts correctly? What this would be for $n=3$? And for a general $n$?
 A: What you have described there are the unit disks in dimensions $1$ and $2$.
The unit tori as described by $\Bbb R^n/\Bbb Z^n$ are not subsets of Euclidean space. They are quotients of Euclidean space. Note that the quotient here is meant algebraically: Two elements are in the same class, not just if they both have integer coordinates, but if their difference has integer coordinates. So for $n = 1$, all integers are in the same class, but $\frac13$ and $-\frac23$ are also in a class together, and so on, for all real numbers.
Thus, for instance, $\Bbb R/\Bbb Z$ means to take the real line and coil it up on itself to a single circle, not to glue together just the integers (and get a bunch of circles all stuck together in a single point). This is, admittedly, a weakness in the notation.
One can, however, still talk about a fundamental cell of a torus: a connected subset of $\Bbb R^n$ that is mapped bijectively (although not homeomorphically) onto the torus.
There are many possible choices for such cells, but perhaps the most conventional is the unit hypercube:
$$
\{(x_1, x_2, \ldots, x_n)\in \Bbb R^n \mid 0\leq x_i< 1 \text{ for all }i\}
$$
For $n = 1$, this becomes $[0, 1)$. For $n = 2$, the standard fundamental cell of the unit torus becomes $[0, 1)\times [0, 1)$, and this pretty obvious pattern of mulitplying together $n$ copies of $[0, 1)$ continues indefinitely.
