Let be a Hausdorff, paracompact topological space (as I am mainly concerned with manifolds) and let be an indexing set that is partially ordered (we will denote the order relations using ).
A locally finite collection of disjoint, locally closed manifolds is called an -decomposition if
satisfies the frontier condition, that respects the partial ordering. That is,
Note that if , then we write .
Similarly, if and then we write .
We can define the dimension of a decomposed space to be
Depth
We can then define the depth of a stratum (piece) as
Said another way, the depth of a piece, is how many other pieces it “fits into” respecting the grading (how many layers of subspaces there are over the piece).
If we look at the cone over , , then it can be broken up into two parts: the “cone part” and the boundary, which is .
Thus,
Stratified space
We can use this to define a stratified space:
A decomposition of is called a stratification (and thus is a stratified space) if the strata satisfy the following condition:
For each , there exists an open neighborhood of , and a stratified space (called the link of ) and a homeomorphism to some subspace containing such that
preserves the decomposition.
In other (less formal) words, inside a stratum, we must find a neighborhood that when “streched” up in dimensions using the cone of (another stratified space) looks just like the entire space of .
So it doesn’t matter what layer you go down, you can always stack stratum “above” it and get a homeomorphism back to the entire space .
Smooth structure
A smooth structure is an additional structure on a stratified space to make each of the stratum smooth manifolds.
Definition
A smooth structure on a stratified space is a subalgebra such that for any then the restriction to any stratum
is smooth.
Note that between strata it need not be smooth, just continuous.