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.)


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.

non Whitney (a) stratification

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 $X$.


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