Application of good coverings to collapsing Alexandrov spaces

Let $M$ be an Alexandrov space collapsing to $X$ of lower dimension. Suppose has no proper extremal subsets and let $F$ denote a regular fiber. We slightly improve the result Perelman construct infinitely long exact sequence homotopy groups spectral cohomology for pair $(M,X,F)$. The proof is application good coverings spaces introduced by Mitsuishi-Yamaguchi. also extend this each primitive subset $X$.