Consider a manifold of dimension $n$, we have two submanifolds $A$ and $B$, with dimension $n_1$ and $n_2$ that satisfies $n_1+n_2=n$. Can we conclude they only intersect at finite points (or isolated points)? How does one prove this?
Motivation: The reason that I asked this question is I was looking at the definition of the intersection number. In wiki it says the following
So I wonder why we have they intersect generically at isolated points when the dimension adds up to the total dimension.
This isn't true even if you restrict yourself to $\mathbb{R}^n$. For example, consider these two 2-dimensional planes in $\mathbb{R^4}$: $$A = \{(x,y,0,0) \mid x,y \in \mathbb{R}\}$$ $$B = \{(x,0,z,0) \mid x,z \in \mathbb{R}\}$$ Then, $A\cap B$ is the line $\{(x,0,0,0)\mid x \in \mathbb{R}\}$.