In *$D$-Modules, Perverse Sheaves and Representation Theory* from R. Hotta, K. Takeuchi and T. Tanisaki, I found the following statement (in section 8.2, the lines before Definition 8.2.2):


Setting: Let $X$ be an analytic space, $U\subset X$ Zariski-open, $j\colon U\hookrightarrow X$ the open embedding, $D^b\_c(U)$ the (complexes of) sheaves with constructible cohomology, and $F\in D^b\_c(U)$. Then there is stated basically the following: 



> 
> If $\sqcup \_{\alpha\in A}X\_\alpha$ is a stratification of $X$ such that $F$ and $\mathrm{D}\_U(F)$ (the dual of $F$) have locally constant cohomologies on each stratum $X\_\alpha$, and one assumes that the given stratification satisfies the Whitney-conditions, then 
>  $Rj\_\ast (F)\vert\_{X\_\alpha}$ and $j\_!(F)\vert\_{X\_\alpha}$ have locally constant cohomologies again, so in particular $Rj\_\ast(F),j\_!(F)\in D^b\_c(X)$. 
> 
> 
> 


Unfortunately, I don't see why this would be obviuos, so my question is:


Could anyone tell me a reference for this? As this is actually the first time I heard of the term "Whitney stratification", I would appreciate very much a reference that goes into some details.


Otherwise, if the statement should be trivial (sorry for my question in this case), any explanation on that would help me out a lot as well.


Thank you very much in advance!



condition for constructibility of direct images of constructible sheaves under open embedding
For a discussion of Whitney stratifications, see for example *Stratified Morse theory* Goresky and Macpherson, *Notes on topological stability* by Mather, or *Stratifications de Whitney et théorème de Bertini-Sard* by Verdier. The point is that the conditions imply that everything is topologically locally trivial along strata in a suitably strong sense; constructibility of direct images follows pretty easily from this. To be a bit more explicit, assume for example, that $F$ is locally constant and that $X-U=X\_\alpha$ (one can reduce to this case). Then locally the inclusion $j:U\to X$ is homeomorphic to $X\_\alpha\times (L\cap U)\subset X\_\alpha \times L$ for a suitable transversal slice $L$. Then $R^ij\_\*F|\_U$ is $F$ when $i=0$ and $0$ otherwise, and $R^i j\_\*F|\_{X\_\alpha}$ is the locally constant sheaf associated to $H^i(L\cap U, F)$, so $Rj\_\*F$ is constructible.


