In topology, a branch of mathematics, an abstract stratified space, or a Thom–Mather stratified space is a topological space X that has been decomposed into pieces called strata; these strata are manifolds and are required to fit together in a certain way. Thom–Mather 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. Basic examples of Thom–Mather stratified spaces include manifolds with boundary (top dimension and codimension 1 boundary) and manifolds with corners (top dimension, codimension 1 boundary, codimension 2 corners), real or complex analytic varieties, or orbit spaces of smooth transformation groups.

Definition

A Thom–Mather stratified space is a triple where V is a topological space (often we require that it is locally compact, Hausdorff, and second countable), is a decomposition of V into strata, and is the set of control data where T_X is an open neighborhood of the stratum X (called the tubular neighborhood), is a continuous retraction, and is a continuous function. These data need to satisfy the following conditions.

Examples

One of the original motivations for stratified spaces were decomposing singular spaces into smooth chunks. For example, given a singular variety X, there is a naturally defined subvariety,, which is the singular locus. This may not be a smooth variety, so taking the iterated singularity locus will eventually give a natural stratification. A simple algebreo-geometric example is the singular hypersurface where is the prime spectrum.