A 0-dimensional stratified pseudomanifold $X$ is a discrete set of points with the trivial filtration $X=X^{0}\supset X^{-1}=\emptyset$. Let $X$ be a compact $n$-dimensional filtered space with the filtration: $$ X=X^{n}\supset X^{n-1}\supset\cdots\supset X^{0}\supset X^{-1}=\emptyset. $$ We say that $X$ is a $n$-dimensional stratified pseudomanifold if it satisfies topological local triviality:

For each point $x$ in the stratum $S_{i}=X^{i}-X^{i-1}$ there exist an open neighborhood $U$ of $x$ in $X$, an open ball $\mathbb{B}^{i}$ of $x$ in $S_{i}$, a compact $(n-i-1)$-dimensional stratified pseudomanifold $L$, and a homeomorphism $$ \phi:\mathbb{B}^{i}\times c(L)\longrightarrow U $$ which takes each $\mathbb{B}^{i}\times c(L^{j-1})$ homeomorphically onto $X^{i+j}\cap U$. Here $L$ is called the link of $x$.

Now assume that Y a closed subset of X, and Y has a filtration induced from the filtration of X, $$ Y=Y^{k}\supset Y^{k-1}\supset\cdots\supset Y^{0}\supset Y^{-1}=\emptyset $$ which means that each stratum of Y is a submanifold of a stratum of X. So my question is: Does Y also satisfy the topological local triviality with respect to the induced filtration?