A remark on the separation by immersions in codimension 1

Let f : Mn−1 → Nn be an immersion with normal crossings from a compact connected (n − 1)-manifold M into a connected, open or compact n-manifold N, where M and N can have boundaries. In this paper, we give a necessary and sufficient condition for f to be an embedding using the number of connected components of N − f(M). We also obtain an estimate from the above for the number of connected components of N − f(M) for f with only double points as its self-intersection points.