In the definition of an $n$-dimensional stratified pseudomanifold one demands the following filtration
$X=X\_n \supset X\_{n-1}=X\_{n-2} \supset X\_{n-3}\supset ... \supset X\_0 \supset X\_{-1}=\emptyset$. where all $X\_i$'s are closed subspaces of X.
Why is it required that $X\_{n-1}=X\_{n-2}$?



Stratified pseudomanifold
I think the reason must be that a pseudomanifold $V$ has singular locus $\Sigma V$ of codimension 2 or greater. (Stratifications of varieties are obtained by letting $X\_{k-1}$ be the singular locus of $X\_k$, or some refinement of that to get the Whitney conditions.) This codimension condition is reflected topologically by the nonbranching condition in the definition of [pseudomanifold](https://encyclopediaofmath.org/wiki/Pseudo-manifold).


