What is $[0,1] \times [0,1]$ I am under the impression that $[0,1) \times [0,1)$ is the torus. Or I was wrong, when we identify o and 1, we also need 1 to be included in the set?
 A: Actually, $[0,1)\times [0,1)$ can be a square, a torus, a set of independent points, or even a sphere. It all depends on what topology you define on it.
Now usually if you name a subset of $\mathbb R^2$ without specifying a specific topology, it is assumed that the topology is simply the standard topology of $\mathbb R^2$. Using that topology, $[0,1)\times [0,1)$ is simply a square.
Now, is it an open set, a closed set, or neither? Well, that of course depends on the chosen topology (this dependence is obvious), but even if you assume the standard topology, it depends also on whether you consider it as the complete set, or as a subset of $\mathbb R^2$. In the first case it is both open and closed, because by the axioms of topology the full set is always open, and so is the empty set, making the full set also closed. However as subset of $\mathbb R^2$, it is neither open nor closed. I don't know if "half-open" is a well-defined concept for two-dimensional objects, but if there's any two-dimensional set for which it makes sense, it's this square.
A: $[0,1) \times [0,1)$ is not a torus, it is a "half-open" square. This is a manifold with corners; its boundary is an open interval with an "L" shape (not smooth at the corner). As you noted, the product of manifold with (smooth) boundaries is not necessarily a manifold with (smooth) boundary, but it's a topological manifold with boundary.
From the second part of your question I guess you were thinking about the torus $T = [0,1]^2 / \sim$ where $(x,0) \sim (x,1)$ and $(0,x) \sim (1,x)$. This space is indeed a torus. It is also true that, is $p : [0,1]^2 \to T$ is the quotient map, the image under $p$ of the space $[0,1)^2$ is the whole torus. But it is abusive to call $[0,1)^2$ the torus; it's its image under $p$ that is a torus.
A: Your "half open square" is a fundamental region of a lattice - with its translates, it fills the real plane. It is the plane modulo the lattice which is the torus. 
The half open square, being a fundamental region, can be identified with the torus, since there is an obvious bijection between points on the torus and points in the square. This bijection can be used to give the half-open square the structure and topology of the torus.
The torus, being closed and bounded, is compact. The half-open square under the natural topology inherited from the real plane is clearly not compact. So they are different.
