Oriented real vector bundles are classified by BSO

Let $(E,o)$ be a numerable oriented bundle. Use [F1] to choose a countable embedding $j:E\to X\times {\mathbb{R}}^{\mathrm{\infty }}$. Give the image plane $j(E_x)$ the orientation transported by $j_x$ from $o_x$. In oriented local frames this varies continuously, so it defines $c_j^{+}:X\to {\mathrm{Gr}}_n^{+}({\mathbb{R}}^{\mathrm{\infty }})$. The tautological pullback map $e\mapsto (p(e),j(e))$ is orientation-preserving by construction. Thus every oriented bundle is in the image.

If two maps to the oriented Grassmannian are homotopic, pull back ${\gamma }_n^{+}$ over $X\times I$ and repeat the graph-transport proof used in [F1] with oriented charts. Every transition matrix has positive determinant, so the endpoint isomorphism is orientation-preserving. Hence pullback depends only on the homotopy class.

Conversely, an orientation-preserving isomorphism between two pullbacks identifies them as one oriented bundle. Move their embeddings to odd and even coordinates and interpolate as in the injectivity proof of [F1], transporting the fixed domain orientation to every intermediate image plane. This is a homotopy through oriented Grassmannian maps, so the pullback assignment is injective.

Pullback of the transported image orientation commutes with base change, proving naturality. By [F2], the same classification can be read as classification of the corresponding $\mathrm{SO}(n)$ reductions, which agrees with the notation in [F3]. When $n=0$, the oriented Grassmannian, structure group, and bundle fiber are points, so both sets are singletons. AC is inherited exactly from [F1]'s numeration and countabilization.