intersection of Whitney stratifications

Question

Let $X$ be an oriented smooth manifold with dimension $n$. If $U$ and $V$ are two oriented closed submanifolds of $X$ and $U$ is transverse to $V$ in $X$. Then $U\cap V$ (suppose the intersection is not empty) is a submanifold of $X$ with dimension $\dim U+\dim V-n$. Now if $X$ is a Whitney stratification and $U,V$ are Whitney substratifications, i.e. $U$, $V$ are subsets of $X$ with Whitney stratification structure, and each stratum of $U$ and $V$ contained in a single stratum of $X$. Assume that $U$ is transverse to $V$ in $X$, i.e. we require the usual transversal conditions on each stratum, then is it true that $U\cap V$ is a Whitney substratification of $X$ with dimension $\dim U + \dim V - n$? Where can I find some reference on this topic?

Answer 1

I wanted to just put a comment but mathoverflow did not allow me.. hope it will soon.

The answer is yes and in fact there are better results.

Check this article:

P. Orro and D. Trotman, 'Regularity of the transverse intersection of two regular stratifications', London Mathematical society, Lecture Note Series 380, Cambridge University Press (2010), p. 298-304.

Answer 2

The Answer is yes. For details see Chap. I, $\S 1$ of

Topological Stability of Smooth Mappings, Lecture Notes in Mathematics, vol. 552, Springer Verlag, 1976.

The answer can be found in Proposition (1.3) Chap.I of the above reference.