Type of topological space

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.

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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 $S_{x))$ 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*, $S_{x))$ is the set germ at *y* of the stratum of the prestratification on *U* containing *y*.

A stratified space is then a topological space equipped with a stratification.

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Pseudomanifold

In the MacPherson's **stratified pseudomanifolds**; the strata are the differences *X*_{i+i}-X_{i} 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 *R*^{n}x 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*.

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.