We denote the n-dimensional real euclidean space as $\mathbb{R}^n$ and the unit sphere in $\mathbb{R}^{n+1}$ as $S^n$.
I have read this example where we show that $\mathbb{R}^n$ is a CW complex by saying that the 0-cells are points with integer coordinates with n-cells being cubes with vertices as these points.
Also, $S^n$ is a CW complex by choosing a point to be 0-cell and the complement of it to be the n-cell.
My question is how do these 'decompostions' make these spaces into CW complexes - don't they need to be defined inductively? What happened to the attaching maps at the levels between 0 and n?
I have seen an inductive 'decomposition' of $S^n$ that clearly shows that it is a CW complex. However, it is the above 'decomposition' that I am not sure about. Also, I have no clue about inductive 'decomposition' of $\mathbb{R}^n$.
Any comments would be helpful.
Thanks.