Hatcher's definition of cell complex - Why is the sphere $S^n$ a cell complex?

Question

I get that the $n$-cell is attached by the constant map $S^{n - 1} \to e^0$, as Hatcher's book says. But from the definition of a cell complex in Hatcher's book, it seems that an $n$-skeleton must contain at least one $n$-cell, so it seems like we are just skipping the $1$-cell through the $n - 1$-cell.

Answer 1

No, the $k$-skeleton needn't contain any $k$-cells. Skipping is allowed, so it is perfectly valid for us to take $X_k$ to be just $X_{k-1}$ without any new cells. This way, to build the $n$-sphere, you take $\{e_0\}=X_0=X_1=X_2=\dots=X_{n-1}$, and only at this point you add a new $n$-cell to get $X_n$ homeomorphic to an $n$-sphere.