I am reading De Cataldo and Migliorini's beautiful paper "the hard Lefschetz theorem and the topology of semismall maps", where they construct an intersection form for each relevant strata, and whose nondegeneracy implies the decomposition theorem for semismall maps.
To be more precise, $f:X\rightarrow Y$ be a semismall maps between projective complex algebraic varieties, and $X$ is nonsingular. Let $Y=\sqcup Y_{i}$ be a Whitney stratification and $f:f^{-1}(Y_{i})\rightarrow Y_{i}$ is a toplogically locally trivial fibration. Let $Y_{i}$ be a relevant strata, i.e $2\dim f^{-1}(y)+\dim Y_{i}=\dim Y$ for each $y\in Y_{i}$, they claim that given $y\in Y_{i}$, it has an open neighbourhood $U$ in $Y$, such that: (a) $Y_{i}\cap U$ is contractible; (b) the restriction map $H^{k}(f^{-1}(U))\rightarrow H^{k}(f^{-1}(y))$ is an isomorphism.
I want to ask why we can always find such an open $U$ such that condition (b) holds?
