Transversally intersecting submanifolds

Definition
Let $$M$$ be a manifold of dimension $$m$$ and $$A,B$$ be two tame submanifolds of $$M$$ of codimensions $$a$$ and $$b$$ respectively. We say that $$A$$ and $$B$$ intersect transversally if their intersection is nonempty, and for every $$p \in M$$, there exists an open set $$U \ni p$$ in $$M$$ and a homeomorphism from $$U$$ to $$\R^m$$, under which $$U \cap A$$ and $$U \cap B$$ get mapped to subspaces $$\R^{m-a}$$ and $$\R^{m-b}$$, and $$U \cap A \cap B$$ gets mapped to $$\R^{m-a-b}$$.

The condition of intersecting transversally implicitly incorporates the condition that each submanifold is individually tame.