MAPS OF RANK m

Let f : X -> Y be a function and let m be an infinite cardinal. Then we say that the rank r(f) of f is <= m if broken vertical bar{y is an element of Y : broken vertical bar f(-1)(y)broken vertical bar > 1}broken vertical bar <= m. If m = N(0) then f is of countable rank. In this paper, we study general properties and find some invariants of m-rank maps. Some close relationships between them and monotone maps are revealed. Monotone maps on surfaces are approximated by countable rank monotone maps if the set of local separating points of the range space has a countable closure. Projective classes of countable rank maps and of fully closed maps are evaluated.