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.)