# Non-example for Whitney (a) stratifications

Given a $$C^1$$ stratification $$\mathscr{S}$$ of a $$C^1$$ manifold $$M$$, we write $$N^\ast \mathscr{S}$$ for the union of conormals to the strata. The stratification is said to be Whitney (a) if $$N^\ast \mathscr{S}$$ is closed. Or equivalently, for strata $$X \subseteq \overline{Y}$$ and points $$x \in X$$ and $$y \in Y$$, as $$y \rightarrow x$$ the tangent space $$T_y Y$$ become arbitrarily close to containing $$T_x X$$ (uniformly over compact subsets of $$X$$).

What's a typical non-example of such a stratification not satisfying Whitney's conditions (a)?

(A non-example for Whitney (b) stratification can be found in this question as well as a non-example for Whitney (a) of pairs of manifolds.)

## 1 Answer

I do not know of a simpler concrete example (as in the case of Whitney (b) condition) of a non-example for Whitney (a). But, a typical non Whitney (a) is as depicted in the picture. Observe that $$X \subset \overline Y$$, but $$Y$$ is not Whitney (a)-regular over $$X$$. Take for example a sequence of points $$\{y_n\}$$ converging to $$x$$ on the edge of the 'turn' of $$Y$$, then the limit of the tangent spaces at $$y_i$$'s does not contain the tangent space at $$x$$ of $$X$$.