stratified topological space

Definition

Decomposition

This definition comes from @sjamaar1991Stratified.

Let $X$ be a Hausdorff, paracompact topological space (as I am mainly concerned with manifolds) and let $I$ be an indexing set that is partially ordered (we will denote the order relations using $\le$ ). A locally finite collection of disjoint, locally closed manifolds $X_i\subset X$ is called an $I$-decomposition if

1. $X={\bigcup }_{i\in I}{X}_i$

2. $\{X_i{\}}_{i\in I}$ satisfies the frontier condition, that respects the partial ordering. That is,

$$X_i\cap \overline{X_j}\ne \varnothing ⟺X_i\subset \overline{X_j}⟺i\le j$$

Note that if $X_i\subset {\overline{X}}_j$, then we write $X_i\le X_j$. Similarly, if $X_i\le X_j$ and $X_i\ne X_j$ then we write $X_i<X_j$.

We can define the dimension of a decomposed space $X$ to be

$$\dim X=\underset{i\in I}{\sup }\dim X_i$$

Depth

We can then define the depth of a stratum (piece) $X_i$ as

$${\text{depth}}_X{X}_i=\sup \{n|\text{there exists pieces}{X}_i<X_{i,1}<\cdots <X_{i,n}\}$$

Said another way, the depth of a piece, is how many other pieces it “fits into” respecting the $I$ grading (how many layers of subspaces there are over the piece).

If we look at the cone over $X$, $\text{cone}(X)$, then it can be broken up into two parts: the “cone part” and the boundary, which is $X$. Thus,

$$\text{depth}(\text{cone}(X))=\text{depth}(X)+1.$$

Stratified space

We can use this to define a stratified space: A decomposition of $X$ is called a stratification (and thus $X$ is a stratified space) if the strata satisfy the following condition: For each $x\in X_i\subset X$, there exists an open neighborhood $U\subset X_i$ of $x$, and a stratified space $L$ (called the link of $x$) and a homeomorphism to some subspace $V\subset X$ containing $x$ such that

$$\phi :U\times \text{cone}(L)⟶V\subset X$$

preserves the decomposition.

Examples

todo