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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 [[Hassler Whitney|Whitney]]. They were introduced by [[René Thom]], who showed that every [[Whitney conditions |Whitney stratified space]] was also a topologically stratified space, with the same strata. Another proof was given by [[John Mather (mathematician)|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==

The definition is inductive on the dimension of ''X''. An ''n''-dimensional '''topological stratification''' of ''X'' is a [[Filtration (mathematics)|filtration]]

:<math> \emptyset = X_{-1} \subset X_0 \subset X_1 \ldots \subset X_n = X </math>

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

:<math> X_i \smallsetminus X_{i-1} </math>,

there exists a neighborhood

:<math> U \subset X </math>

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

:<math> U \cong \mathbb{R}^i \times CL</math>.

Here <math>CL</math> is the open [[Cone (topology)|cone]] on ''L''.

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

:<math> X_i \smallsetminus X_{i-1} </math>.

Connected components of ''X<sub>i</sub> \ X<sub>i-1</sub>'' are also frequently called strata.

==See also==
* [[Singularity theory]]
* [[Whitney conditions]]
* [[Stratifold]]
* [[Intersection homology]]

==References==
* [[Mark Goresky|Goresky, Mark]]; [[Robert MacPherson (mathematician)|MacPherson, Robert]] ''Stratified Morse theory'', Springer-Verlag, Berlin, 1988.
* [[Mark Goresky|Goresky, Mark]]; [[Robert MacPherson (mathematician)|MacPherson, Robert]] ''Intersection homology II'', Invent. Math. 72 (1983), no. 1, 77--129.
* [[John Mather (mathematician)|Mather, J.]] ''[http://www.math.princeton.edu/facultypapers/mather/notes_on_topological_stability.pdf Notes on topological stability]'', Harvard University, 1970.
* [[René Thom|Thom, R.]] ''[http://www.ams.org/bull/1969-75-02/S0002-9904-1969-12138-5/ Ensembles et morphismes stratifiés]'', Bulletin of the American Mathematical Society 75 (1969), pp.240-284.
* {{cite book
|last=Weinberger
|first=Shmuel
|author-link=Shmuel Weinberger
|title=The topological classification of stratified spaces
|series=Chicago Lectures in Mathematics
|publisher=University of Chicago Press
|location=Chicago, IL
|year=1994
|url=http://www.press.uchicago.edu/ucp/books/book/chicago/T/bo3625837.html
|isbn=9780226885667
}}

[[Category:Singularity theory]]
[[Category:Generalized manifolds]]

{{topology-stub}}

Ruppeiner geometry - Revision history

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