CW Structure of $SU$

Question

I'm reading Switzer's Algebraic Topology and he mentions that $SU = SU(\infty)$ can be given a CW complex structure. He also says that this implies, by a theorem of Milnor's, that $\Omega SU$ has the homotopy type of a CW complex. What is the CW structure on $SU$, and where can I find a proof of Milnor's theorem?

Answer 1

In On the Cell structures of SU(n) and Sp(n), Ichiro Yokata gives a cell structure of $SU(n)$ such that $SU(n-1)$ (with the cell structure he gives) is a subcomplex. Then one can obtain a cell structure on $SU$ by taking the limit of these.

The result you quote is in Milnor's On spaces having the homotopy type of a CW complex: if $X$ is any space with the homotopy type of a CW complex, so is $\Omega X$. This follows from his Theorem 3 by taking $C = S^1$.