---
tag: P9R3777ZJF
kind: concept
name: Thom–Mather stratified space
msc: [58A35, 57N80, 32S60]
sources:
  - url: https://en.wikipedia.org/wiki/Thom%E2%80%93Mather_stratified_space
same_as:
  wikidata: Q7825045
  wikipedia: Thom–Mather stratified space
---

## Summary

A Thom–Mather stratified space is a space cut into smooth manifolds, the strata, together with control data near each stratum.
The control data are a tubular neighbourhood, a retraction onto the stratum and a distance function to it.
The control data must be compatible between strata.
Mather showed that every closed [[Whitney stratified|3F84AMVJNV]] subset of a smooth manifold carries such data.
The control data are what makes the [[Thom–Mather isotopy lemmas|7G1PCS2RPG]] work, so these spaces are locally topologically trivial along each stratum.

## Explanations

### L1 human

Think of a cone: it has a tip and a smooth part, two pieces of different dimension.
Near the tip one can measure the distance to the tip and project nearby points to it.
A Thom–Mather structure asks for such a projection and such a distance near every piece, and asks the projections and distances of different pieces to agree with each other.
With these data one can build vector fields that respect all the pieces, and flow along them.
This is how one proves that the space looks the same at all points of a stratum.