Separation by a codimension-1 map with a normal crossing point

Let f:M^n-1→Nⁿ be an immersion with normal crossings of a closed orientable (n-1)-manifold into an orientable n-manifold. We show, under a certain homological condition, that if f has a multiple point of multiplicity m, then the number of connected components of N-f(M) is greater than or equal to m+1, generalizing results of Biasi and Romero Fuster (Illinois J. Math.36 (1992), 500-504) and Biasi, Motta and Saeki (Topology Appl.52 (1993), 81-87). In fact, this result holds more generally for every codimension-1 continuous map with a normal crossing point of multiplicity m. We also give various geometrical applications of this theorem, among which is an application to the topology of generic space curves.