Subordinate flag for a family of linear isometries

Given a compact parameter space $X$ and a family $f_x:U\to U\prime$ of linear isometries, a subordinate flag for a family of linear isometries consists of nested cofinal source and target subspaces $V_i,W_i$ with $f_x(V_i)\subset W_i$ for every $x$. After enlarging the target stages, compactness makes this possible for every finite source stage. It permits simultaneous finite-dimensional Thom space constructions.