# intersection of Whitney stratifications

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?

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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.

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.

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