Union of two manifolds of different dimension is still a manifold?

Question

I know the union of two manifolds, let’s say of dimension k both, is a manifold iff in every point p of the intersection, you can find an open neighborhood in the union such that it looks like (an open subset of) R^{k}. But, how would this work for two manifolds M and N of dimensions m and n respectively? M and N are not disjoint necessarily.