It is known that transversality of submanifolds is generic in the sense that two submanifolds could be made transversal by small perturbations. I was wondering if the same is true for subbundles of $TM$. Let $\Delta_1$ be a 1-dimensional smooth subbundle of $TM$ and $\Delta_2$ a codimension 1 subbundle of $TM$. Is it true that in every arbitrarily small $C^{\infty}$ neighbourhood of $\Delta_2$, there exists a distribution $\Delta_{2,\epsilon}$ transverse to $\Delta_1$? I feel like it should go like this: transversality is an open condition so the set of points where the distributions are non-transversal is a closed subset of $M$. It seems like one should do small perturbations around these points, perhaps using partitions of unity. However closed subset can be very bad, fractal for instance and I am not sure how to carry out such arguements when the set in question is not a closed submanifold.
So any answer or reference to usage of such techniques is very welcome. Thanks