Let $M,N \subset Y$ be sub manifolds. If $x \in M\cap N$, then $T_xM + T_xN = T_xY$ means $M$ and $N$ are transversal. But if $M \cap N = \emptyset$, then the results hold vacuously?
So doesn't that imply $T_xM$ and $T_xN$ are complements even though no assumption on the tangent spaces were made? I must be missing something here...
Two submanifolds $M$ and $N$ of $Y$ are said to intersect transversally if for every $x\in M\cap N$ we have $T_xM+T_xN = T_yY$. This is of course vacuously true if $M\cap N = \emptyset$. It doesn't even make sense to talk about $T_xM$ nor $T_xN$ if we are considering $x\in M\cap N=\emptyset$.