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-set. 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.

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.

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

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

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-set. 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.