Example 25.5 (Real projective spaces):

We take the space $X$ to be the closed unit disc $E^n$ in $\mathbb{R}^n$ and $A$ as its boundary. The space $B$ is the lower dimensional projective space $\mathbb{R}P^{n-1}$. The attaching map is the quotient map $p:S^{n-1}\longrightarrow \mathbb{R}P^{n-1}$. We leave the proof of the following result to the reader.