---
tag: F19BQTN832
kind: definition
name: Whitney stratification
msc: [32S60, 58A35, 57N80]
defines: 3F84AMVJNV
uses: [VKFW27WXEZ, WY6K3P4YCG]
sources:
  - url: https://en.wikipedia.org/wiki/Whitney_conditions
---

## Definition

Let $X$ be a closed subset of $\R^N$.
Let $S$ and $S'$ be disjoint smooth [[submanifolds|VKFW27WXEZ]] of $\R^N$ and let $y \in S'$.
The pair $(S, S')$ satisfies *Whitney's condition (b)* at $y$ if the following holds.
Take any sequences $x_k \in S$ and $y_k \in S'$ that both converge to $y$, with $x_k \ne y_k$.
If the [[tangent spaces|WY6K3P4YCG]] $T_{x_k} S$ converge to a subspace $\tau$ and the lines through $x_k$ and $y_k$ converge to a line $\ell$, then $\ell \subseteq \tau$.
The pair satisfies *condition (a)* at $y$ if the following holds.
For every sequence $x_k \in S$ converging to $y$ such that $T_{x_k} S$ converges to a subspace $\tau$, one has $T_y S' \subseteq \tau$.

A *Whitney stratification* of $X$ is a locally finite partition of $X$ into connected smooth submanifolds of $\R^N$, called *strata*, such that:

1. each stratum is locally closed in $X$;
2. if a stratum $S'$ meets the closure $\overline{S}$ of a stratum $S \ne S'$, then $S' \subseteq \overline{S} \setminus S$ (the frontier condition);
3. every such pair $(S, S')$ satisfies condition (b) at every point of $S'$.

## Remarks

Condition (b) implies condition (a).
The same definition applies to subsets of $\C^N = \R^{2N}$, and for complex analytic sets one asks the strata to be complex submanifolds.

Example: the cone $X = \set{x^2 + y^2 = z^2} \subset \R^3$ with three strata: $\set{0}$ and the two connected components of $X \setminus \set{0}$, where $z > 0$ and $z < 0$.
The frontier condition holds, and condition (b) holds at $0$.
Indeed, the cone is a union of lines through $0$, so the line from $0$ to $x_k$ lies in the tangent plane $T_{x_k} X$, hence also in the limit plane.

Non-example: the complex Whitney umbrella $X = \set{x^2 = z y^2} \subset \C^3$ with the two strata $S'$, the $z$-axis, and $S = X \setminus S'$.
The surface $S$ is smooth, since the gradient $(2x, -2zy, -y^2)$ of $x^2 - z y^2$ is nonzero off the $z$-axis.
Every point $(0, 0, c)$ is a limit of points $(\sqrt{c}\, t, t, c)$ of $S$ as $t \to 0$, so the frontier condition holds.
For small real $s > 0$, the point $x_s = (s^{3/2}, s^{1/2}, s^2)$ lies on $S$ and tends to $0$.
There the normal vector is $(2 s^{3/2}, -2 s^{5/2}, -s)$, which, divided by $s$, tends to $(0, 0, -1)$.
So $T_{x_s} S$ tends to the complex plane $z = 0$, which does not contain $T_0 S'$, the $z$-axis.
Condition (a) fails at $0$, so condition (b) fails too.
The origin must be made a stratum of its own: the strata $S$, the $z$-axis minus $0$, and $\set{0}$ form a Whitney stratification.