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 Mark Goresky and Robert MacPherson.

[edit] Definition

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 - X_i-1, there exists a neighborhood $U \subset X$ of x in X, a compact i-1-dimensional stratified space L, and a filtration-preserving homeomorphism $U \cong \mathbb{R}^i \times CL$ . Here $C L$ is the open cone on L.

If X is a topologically stratified space, the i-dimensional stratum of X is the space X_i - X_i-1. Connected components of X_i - X_i-1 are also frequently called strata.