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The notion of cs-stratification of a topological space is apparently due to Siebenmann, see also the paper by N. Habegger and L. Saper in the paper "Intersection cohomology of cs-spaces and Zeeman's filtration", Invent.Math. (1991).

First I remind its definition and then formulate my question.

Definition. Let $X$ be a topological space. A filtration $\mathcal{X}$ of $X$ by closed subsets $$X=X_n \supset X_{n-1}\supset\dots \supset X_0\supset X_{-1}=\emptyset$$ is called cs-stratification of dimension $n$ if it satisfies the following properties:

(i) The stratum $S_{n-k}:=X_{n-k}\backslash X_{n-k-1}$ is either empty or a $(n-k)$-dimensional topological manifold.

(ii) $S_n=X_n\backslash X_{n-1}$ is non-empty.

(iii) (Local Normal Triviality) For any $x\in S_{n-k}$ there exists a neighborhood $U$ of $x$ in $X$ and a homeomorphism \begin{eqnarray}\label{E:1} U=B\times c(L),\,\,\, (\ast) \end{eqnarray} where $B$ is an open $(n-k)$-dimensional ball, and $c(L)$ is the open cone \begin{eqnarray*} c(L)=L\times [0,1)/(x,0)\sim(x',0),\\ (c(\emptyset)\mbox{ is a point}) \end{eqnarray*} on a compact topological space $L$ (called the link). $L$ is assumed to have a filtration $\mathcal{L}$ $$L=L_{k-1}\supset L_{k-2}\supset \dots\supset L_0\supset L_{-1}=\emptyset$$ for which the homeomorphism ($\ast$) satisfies $$U\cap X_{n-j}=B\times c(L_{k-j-1}).$$

As far as I understand it follows automatically that any subset $L_k\subset L$ is closed. But my question is as follows.

Question. It is not explicitly mentioned in the definition that the filtration $\mathcal{L}$ of $L$ is a cs-stratification (by induction on the dimension). Was this assumption omitted by purpose or it should be added? Does it follow automatically from the other assumptions?

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You should look at Greg Friedman's book: http://faculty.tcu.edu/gfriedman/IHbook.pdf CS-sets are discussed in section 2.3. In fact for many purposes it is rather interesting to suppose that the links $L$ are just compact filtered topological spaces and not CS-sets. Deligne's axioms for intersection homology sheaves are easy to verify in this setting where we just suppose that links are compact.

The definition does not imply $L$ is itself a CS-set.

Such a recursive assumption is useful when you want to study pseudomanifolds and Poincaré duality of pseudomanifolds in intersection homology.

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    $\begingroup$ In Siebenmann's "Deformation of Homeomorphisms on Stratified Sets," where he introduces CS sets, he explicitly notes on page 128 that he does not make this assumption. You can still prove a surprising amount about these spaces without such an assumption! (And thanks, David, for referencing my book!) $\endgroup$
    – Greg Friedman
    Commented Jul 30, 2016 at 7:26

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