Real and complex vector bundles are classified by stable Grassmannians

By [F6], the stable Grassmannian is paracompact. Apply [A1] and [F7] to the linear graph-chart cover of ${\gamma }_n$; this supplies a numeration of the tautological bundle, and every pullback of that numeration is again a numeration. Thus $\mathrm{\Phi }([f])=[f^{\ast }{\gamma }_n]$ always lands in numerable bundles.

If $f_0\simeq f_1$ and $X$ is paracompact, [F2] makes the two pullbacks isomorphic. For arbitrary CGWH $X$, pull back the numerable principal Stiefel bundle along the homotopy. The equivariant endpoint-transport argument in [F4] applies to this numerable principal $\mathrm{O}(n)$- or $\mathrm{U}(n)$-bundle and is a principal-bundle isomorphism; [F3] makes the associated vector bundles isomorphic. Hence $\mathrm{\Phi }$ is well-defined in both stated branches.

Let $E\to X$ be numerable. By [F1] it has an embedding $j:E\to X\times {\mathbb{F}}^{\mathrm{\infty }}$, and its image-plane map $c_j$ satisfies $c_j^{\ast }{\gamma }_n\cong E$. Thus $\mathrm{\Phi }$ is surjective. The paracompact branch obtains a numeration from [F7], while the explicitly numerable branch starts with that data; [F1] uses AC to countabilize it.

If $\mathrm{\Phi }([f_0])=\mathrm{\Phi }([f_1])$, identify the two pullbacks. The reverse construction in [F2] moves the two tautological embeddings to odd and even coordinates and interpolates their image planes through fiberwise injections; that construction uses no paracompactness. It yields $f_0\simeq f_1$, so $\mathrm{\Phi }$ is injective. Together with step 3.1 this proves both displayed bijections.

For $a:X^{\mathrm{\prime }}\to X$, the canonical pullback comparison identifies $a^{\ast }(f^{\ast }{\gamma }_n)$ with $(fa{)}^{\ast }{\gamma }_n$, so the bijections are natural. The numerable principal $\mathrm{O}(n)$- or $\mathrm{U}(n)$-bundle $V_n({\mathbb{F}}^{\mathrm{\infty }})\to {\mathrm{Gr}}_n({\mathbb{F}}^{\mathrm{\infty }})$ has contractible total space by [F5], so its quotient is the concrete $B\mathrm{O}(n)$ or $B\mathrm{U}(n)$ model. The published theorem [F4] has the same numerability boundary and likewise excludes nonnumerable bundles.

When $n=0$, the Grassmannian is a point and the only rank-zero bundle is $X\to X$, so both sides are singletons. When $X=\mathrm{\varnothing }$, there is one map and one empty bundle in every rank. AC is used in steps 1.1–3.1 for partitions, endpoint transport, and countabilization; no claim is made for a locally trivial bundle lacking a numeration.