This article relies largely or entirely on a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources.Find sources: "Stratified space" – news · newspapers · books · scholar · JSTOR (May 2024)
This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (May 2024) (Learn how and when to remove this message)

In mathematics, especially in topology, a stratified space is a topological space that admits or is equipped with a stratification, a decomposition into subspaces, which are nice in some sense (e.g., smooth or flat[1]).

A basic example is a subset of a smooth manifold that admits a Whitney stratification. But there is also an abstract stratified space such as a Thom–Mather stratified space.

On a stratified space, a constructible sheaf can be defined as a sheaf that is locally constant on each stratum.

Among the several ideals, Grothendieck's Esquisse d’un programme considers (or proposes) a stratified space with what he calls the tame topology.

A stratified space in the sense of Mather

Mather gives the following definition of a stratified space. A prestratification on a topological space X is a partition of X into subsets (called strata) such that (a) each stratum is locally closed, (b) it is locally finite and (c) (axiom of frontier) if two strata A, B are such that the closure of A intersects B, then B lies in the closure of A. A stratification on X is a rule that assigns to a point x in X a set germ at x of a closed subset of X that satisfies the following axiom: for each point x in X, there exists a neighborhood U of x and a prestratification of U such that for each y in U, is the set germ at y of the stratum of the prestratification on U containing y.[citation needed]

A stratified space is then a topological space equipped with a stratification.[citation needed]


Main article: Pseudomanifold

In the MacPherson's stratified pseudomanifolds; the strata are the differences Xi+i-Xi between sets in the filtration. There is also a local conical condition; there must be an almost smooth atlas where locally each little open set looks like the product of two factors Rnx c(L); a euclidean factor and the topological cone of a space L. Classically, here is the point where the definitions turns to be obscure, since L is asked to be a stratified pseudomanifold. The logical problem is avoided by an inductive trick which makes different the objects L and X.[citation needed]

The changes of charts or cocycles have no conditions in the MacPherson's original context. Pflaum asks them to be smooth, while in the Thom-Mather context they must preserve the above decomposition, they have to be smooth in the Euclidean factor and preserve the conical radium.[citation needed]

See also



Further reading