In topology, a branch of mathematics, a topologically stratified space is a space X that has been decomposed into pieces called strata; these strata are topological manifolds and are required to fit together in a certain way. Topologically stratified spaces provide a purely topological setting for the study of singularities analogous to the more differential-geometric theory of Whitney. They were introduced by René Thom, who showed that every Whitney stratified space was also a topologically stratified space, with the same strata. Another proof was given by John Mather in 1970, inspired by Thom's proof.
Contents
Basic examples of stratified spaces include manifold with boundary (top dimension and codimension 1 boundary) and manifold with corners (top dimension, codimension 1 boundary, codimension 2 corners).
Definition
The definition is inductive on the dimension of X. An n-dimensional topological stratification of X is a filtration
of X by closed subspaces such that for each i and for each point x of
there exists a neighborhood
of x in X, a compact (n - i - 1)-dimensional stratified space L, and a filtration-preserving homeomorphism
Here
If X is a topologically stratified space, the i-dimensional stratum of X is the space
Connected components of Xi Xi-1 are also frequently called strata.
Examples
One of the original motivations for stratified spaces were decomposing singular spaces into smooth chunks. For example, given a singular variety
where