Infinite complex projective space is K(Z,2)

By [F1, F2], the finite stages of Milnor's bundle are the Hopf bundles $S^{2N+1}\to {\mathbb{C}\mathbb{P}}^N$. Passing through their compatible inclusions identifies the infinite bundle with

Assume [A1]. By [F4], the numerable bundle in Step 1.1 is a Hurewicz fibration; the exact AC expenditure is the well-ordering used in that supplier's lifting-function construction. The long exact sequence and contractibility give ${\pi }_k({\mathbb{C}\mathbb{P}}^{\mathrm{\infty }})\cong {\pi }_{k-1}(S^1)$ for $k\ge 2$, while its component segment gives ${\pi }_1({\mathbb{C}\mathbb{P}}^{\mathrm{\infty }})=0$. Since ${\pi }_1(S^1)\cong \mathbb{Z}$ and ${\pi }_j(S^1)=0$ for $j>1$, the sole positive homotopy group is ${\pi }_2\cong \mathbb{Z}$. The space is connected, so a CW model is $K(\mathbb{Z},2)$. $\mathrm{\square }$