Subset cover of a framed vector bundle ($X_k\cong X\times \left(\genfrac{}{}{0ex}{}{\{1,\dots ,n\}}{k}\right)$)

A global ordered framing selects the displayed $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$-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.