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Let $M$ be a smooth manifold. If $X$ and $Y$ are two Whitney objects, i.e. subsets with a given Whitney stratification, then $X$ and $Y$ are transverse if each stratum of $X$ is transverse to each stratum of $Y$. This definition confuses me: if $S\subset X$ is a stratum and $R\subset Y$ is another stratum, and $$\dim S+\dim R < \dim M$$ then how to define the transversal intersection of $S$ and $R$ in $M$?

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    $\begingroup$ $R \cap S = \emptyset$ $\endgroup$ Commented Jul 29, 2013 at 3:09
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    $\begingroup$ Your question is not specific to stratifications. Vivek Shende's comment says it all, but let me add an example: two transverse curves in $3$-space are disjoint curve, and this is easily seen to be a generic property. $\endgroup$ Commented Jul 29, 2013 at 6:36

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