Attaching cells to CW complex yields a CW complex

Question

On page 352 of Hatcher's book, he claims that if we have a CW complex A, then attach some cells to it, what remains is a CW complex. The way he states it, it seems like this should be obvious. But given CW complexes are defined inductively, by attaching n cells to the n-1 skeleton, Hatcher's claim as stated is not obvious to me (he states no conditions on how the cells are attached to A). Is it obvious?

Answer 1

If $A$ is a CW-complex and $\varphi:S^{k-1}\to A$ is a cellular map, then the space $A\cup_\varphi e^k$ obtained by attaching a map via $\varphi$ is a CW-complex. The condition that $\varphi$ is cellular means that its image is contained in the $(k-1)$-skeleton $A^{k-1}$. So actually, when building up $A\cup_{\varphi} e^k$ inductively as a cell complex, you could have added the new $k$-cell when you only had the $(k-1)$-skeleton, instead of adding it at the very end. This is why Hatcher specifies that the maps $\varphi_\alpha$ should be chosen to be cellular maps.