Subset cover of a framed vector bundle
= Subset cover of a framed vector bundle
{title2=$X_k\cong X\times\binom{\{1,\ldots,n\}}k$}
A global ordered framing selects the displayed $\binom nk$-sheeted cover inside the <unordered Stiefel bundle>. It is a trivial cover because the frame labels are global. Selecting subsets of a direct-sum framing decomposes the cover as the disjoint union of fiber products of the two summands' subset covers.