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 subset of a smooth manifold carries such data. The control data are what makes the Thom–Mather isotopy lemmas work, so these spaces are locally topologically trivial along each stratum.