Topologically stratified space

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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.

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[edit]

The definition is inductive on the dimension of X. An n-dimensional topological stratification of X is a filtration

 \emptyset = X_{-1} \subset X_0 \subset X_1 \ldots \subset X_n = X

of X by closed subspaces such that for each i and for each point x of

 X_i \smallsetminus X_{i-1} ,

there exists a neighborhood

 U \subset X

of x in X, a compact n-i-1-dimensional stratified space L, and a filtration-preserving homeomorphism

 U \cong \mathbb{R}^i \times CL.

Here CL is the open cone on L.

If X is a topologically stratified space, the i-dimensional stratum of X is the space

 X_i \smallsetminus X_{i-1} .

Connected components of Xi \ Xi-1 are also frequently called strata.

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