= Classification of vector bundles by a universal quotient bundle
{title2=$E\cong\phi^*Q,\quad\phi:M\to\operatorname{Gr}_{N-d}(\mathbb R^N)$}
On a compact base, finitely many global sections can be selected to span every fiber when evaluations of global sections are surjective. They give a surjective morphism $M\times\mathbb R^N\to E$. Its kernel is a <kernel bundle of a surjective vector bundle morphism>, and its fibers give the displayed map $\phi$. The first isomorphism theorem on each fiber identifies the resulting pullback quotient with $E$, with continuously varying inverses. The finite selection follows by choosing a frame's values at each point, extending each value to a global section, then taking a finite cover of the regions where those sections remain independent.
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