Vector-bundle classification can fail without numerability

The tangent projection $TL\to L$ is a locally trivial rank-one real vector bundle: its linear charts are the derivatives of the smooth coordinate charts on the one-manifold $L$. The base is CGWH and fails only the library's separate second-countability convention, by [F1]. Thus $TL$ satisfies exactly the topological hypotheses in the false claim.

Suppose $TL$ were numerable. By [F2] it would have a continuous positive-definite fiber inner product $g$. We show directly that such a $g$ metrizes $L$. Any two points of the connected one-manifold $L$ can be joined by a piecewise smooth path: the points reachable from a fixed point by finite chains of coordinate intervals form a nonempty open-and-closed set. Define $d_g(p,q)$ as the infimum of the $g$-lengths of these paths. It is finite, symmetric, and satisfies the triangle inequality.

To prove positivity and identify the topology, fix $p\mathrm{\ne }q$ and choose a coordinate interval $U$ about $p$, not containing $q$, with a smaller closed coordinate interval $K\subset U$ whose interior contains $p$. In the coordinate $t$, write $g=a(t)d{t}^2$. On compact $K$, continuity and positivity give $0<m\le a\le M$. Every path from $p$ to $q$ must first leave $K$, so its coordinate variation before leaving is at least the positive coordinate distance from $p$ to $\mathrm{\partial }K$; its length is therefore bounded below by that distance times $\sqrt{m}$. Thus $d_g(p,q)>0$. Conversely, within a still smaller coordinate interval, straight coordinate segments have length at most $\sqrt{M}$ times their coordinate displacement, while the preceding lower bound forces sufficiently small $d_g$-balls to stay in any prescribed coordinate neighborhood. Hence $d_g$ induces exactly the topology of $L$, contradicting the nonmetrizability in [F1]. Therefore $TL$ is not numerable. The implication from a supplied numeration to $g$, and this metric-topology argument, make no choices beyond the supplied data.

By [F3], AC supplies a numeration of the tautological line ${\gamma }_1\to {\mathrm{Gr}}_1({\mathbb{R}}^{\mathrm{\infty }})$, and pulling this fixed numeration back along any map $f:L\to {\mathrm{Gr}}_1({\mathbb{R}}^{\mathrm{\infty }})$ gives a numeration of $f^{\ast }{\gamma }_1$. Therefore $TL\cong f^{\ast }{\gamma }_1$ would contradict step 2.1. No such classifying map exists. This is the sole nonlocal use of AC, recorded by [A1]; the contradiction after a hypothetical numeration is choice-free.

Consequently the locally trivial rank-one bundle $TL$ over the CGWH space $L$ is the required witness, and the false claim fails precisely because it omitted numerability.