I have a question in seeing this $$U(n)=\frac{U(n)}{U(n-1)} * \frac{U(n-1)}{U(n-2)}*\cdots *\frac{U(2)}{U(1)}*U(1)$$

So, group U(n) is written as product of quotient spaces.

Is quotient space, for example $\frac{U(n)}{U(n-1)}$ , as topological space the same as quotient gropup i.e. set of cosets?

How to prove $\frac{U(n)}{U(n-1)}$ is diffeomorphic to some sphere ? Thx