Thom–Mather stratified space

Definition

Let $V$ be a Hausdorff, locally compact topological space with a countable basis of open sets. A stratification of $V$ is a locally finite partition $\mathcal{S}$ of $V$ into locally closed subsets, called strata. Each stratum must be a smooth manifold in the subspace topology. The frontier condition must hold: if $X, Y \in \mathcal{S}$ and $X \cap \overline{Y} \neq \emptyset$, then $X \subseteq \overline{Y}$. One writes $X < Y$ when $X \ne Y$ and $X \subseteq \overline{Y}$. Control data for $\mathcal{S}$ consist of three things for each stratum $X$. They are an open neighbourhood $T_X$ of $X$ in $V$, a continuous retraction $\pi_X \colon T_X \to X$, and a continuous function $\rho_X \colon T_X \to [0, \infty)$ with $\rho_X^{-1}(0) = X$. A Thom–Mather stratified space is a triple $(V, \mathcal{S}, \mathcal{J})$, where $\mathcal{J}$ is control data for $\mathcal{S}$, such that:

1. For all strata $X < Y$, the map $(\pi_X, \rho_X) \colon T_X \cap Y \to X \times (0, \infty)$ is a smooth submersion.
2. For all strata $X < Y$ and every stratum $Z$, the control conditions $\pi_X(\pi_Y(v)) = \pi_X(v)$ and $\rho_X(\pi_Y(v)) = \rho_X(v)$ hold for all $v \in T_X \cap T_Y \cap Z$ for which both sides are defined.