Is $[0,1]\times [0,1]$ homeomorphic to $[0,1]\times [0,1]\times [0,1]$? 
Is $[0,1]\times [0,1]$ homeomorphic to $[0,1]\times [0,1]\times [0,1]$ ?

The properties that  I learnt i.e. compactness,connectedness etc don't allow me to answer the question .Please help.
I have only learnt point set topology.Is it possible to answer this question using that only.
 A: It may turn out that the easiest way to prove $[0,1]^2$ and $[0,1]^3$ aren't homeomorphic is to note


*

*For any point $p \in [0,1]^3$, the space $[0,1]^3 \setminus \{p\}$ is simply connected.

*The space $[0,1]^2 \setminus \{p\}$ is not simply connected for any point in the interior of the square.


Point 1 is easy to prove. Unforunately, point 2 takes a bit more than just the basic ideas of point set topology.
Another potential strategy: note that the "boundary" of $[0,1]^2$ is a 1d square, whereas the "boundary" of $[0,1]^3$ is a 2d cube. We can tell the difference between a 1d square and a 2d cube by noting that the removal of two points can disconnect the former, but not the latter. 
Unforunately, it is not clear how to characterize topologically the points on the boundary. For $[0,1]^2$, we can characterize the boundary points as the points $p$ such that $[0,1]^2 \setminus \{p\}$ is simply connected, but this again uses ideas beyond basic point set topology. Also, if simply connectedness were OK to be used, then we could simply use the first proof outlined above.
