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 $\mathbb {R} ^{m}$, under which $U\cap A$ and $U\cap B$ get mapped to subspaces $\mathbb {R} ^{m-a}$ and $\mathbb {R} ^{m-b}$, and $U\cap A\cap B$ gets mapped to $\mathbb {R} ^{m-a-b}$.

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