# Heredity of stratified pseudomanifolds

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?

• Where's the 'hereditary'? Commented Feb 3, 2021 at 14:55
• My statement may be a little confused. As a stratified pseudomanifold, X is locally cone-like. I want to know if Y is also locally cone-like. Commented Feb 4, 2021 at 1:22
• Is it possible to copy the relevant definitions here? Questions should, ideally, be self-contained. Commented Feb 5, 2021 at 3:48
• Thanks for the reminding. Commented Feb 6, 2021 at 5:40

## 1 Answer

I'm not completely sure I understand the question, but if the only assumption is that $$Y$$ is an arbitrary closed subset of $$X$$ then this will not be true. For example, let $$X$$ be the real line filtered as $$\{0\}\subset \mathbb{R}$$ and let $$Y=\{0\}\cup\{1/n\mid n\in \mathbb Z^+\}.$$ It also will not generally be true that each stratum of $$Y$$ is a submanifold of a stratum of $$X$$. As a simple example, let $$X$$ be a manifold with the trivial filtration and let $$Y$$ be a closed subset that is not a manifold.

In general, closed subsets of stratified pseudomanifolds need not have any particularly nice properties. Though one special case that may be of interest to you are the normally nonsingular inclusions. Those act like "slices" of pseudomanifolds and, in particular, are pseudomanifolds themselves.