Unordered Stiefel bundle
= Unordered Stiefel bundle
For a rank-$n$ Euclidean <vector bundle>, take its ordered orthonormal $k$-frame bundle and quotient by the symmetric group permuting the $k$ vectors. The whole fiber is generally positive-dimensional. A prescribed global framing selects a finite subcover consisting of the unordered $k$-element subsets of its distinguished frame vectors.