Two CW complexes with the same number of $n$-cells for each $n$ homeomorphic?

Question

i am trying to understand Theorem 5.20 in Lee's Introduction to Topological manifolds

I am interested in the uniqueness part. Does this mean that two CW complexes with the same number of $n$-cells for each $n \geq 0$ are homeomorphic?

Answer 1

The key thing here is the data of the attaching maps. Both $S^1\vee S^2$ and $\Bbb RP^2$ have cell structures with one $0$-cell, one $1$-cell, and one $2$-cell. If we attach the $2$-cell with the trivial map, we get $S^1\vee S^2$. If we attach with the antipodal map $S^1\to S^1$, we get $\Bbb RP^2$.