TITLE: Is $\prod \limits_{i = 1}^{n} [0,1] \subseteq \mathbb R^n$ homeomorphic to the closed unit ball?
QUESTION [5 upvotes]: Is $\prod \limits_{i = 1}^{n} [0,1] \subseteq \mathbb R^n$ homeomorphic to $\bar B(\theta , 1)$ , the closed ball centered at origin with radius $1$? 
Can someone please give some reference links to study elementary techniques to deal with homeomorphism and to show two given spaces homeomorphic ?

REPLY [3 votes]: First translate and scale the cube so that it is centered at the origin with side length 2. This map is a homeomorphism.
Now consider the following map from this new cube $C$, to the unit closed ball $B$.
$$ f \colon C \to B $$
$$ f(x) = \frac{|x|}{d_x}x $$
where $d_x$ is defined to be the distance between the origin and the point where the line segment from the origin through $x$ intersects $C$. This map is continuous with continuous inverse and so is a homeomorphism.
Hence as the composition of two homeomorphisms is a homeomorphism we have your desired result.