$\mathbb{C}P^n$-bundle over $BU(n)$

Question

The inclusion $U(1)\times U(n-1)\to U(n)$ induces a map $BU(1)\times BU(n-1)\to BU(n)$. This source claims that the fiber of this map is $\mathbb{C}P^n$. Could someone please explain why this is true?

Answer 1

There is a typo --- it should be $\mathbb{C}P^{n-1}$ (as you can also see from the cohomology ring in the next sentence).

$BU(1)\times BU(n-1)\to BU(n)$ is the classifying map for taking orthogonal direct sum of a rank-$1$ bundle and a rank-$(n-1)$ bundle. So the fibers are ways to split an $n$-dimensional vector space into the sum of $1$- and $(n-1)$-dimensional subspaces, but that is just choosing $1$-dimensional subspaces from a fixed $n$-dimensionsional one (the $n-1$ must then be its orthogonal complement), which is $\mathbb{C}P^{n-1}$.