Lelek maps and n-dimensional maps from compacta to polyhedra

摘要

For a natural number m ≥ 0, a map f: X → Y from a compactum X to a metric space Y is an m-dimensional Lelek map if the union of all non-trivial continua contained in the fibers of f is of dimension ≤ m. In [M. Levin, Certain finite-dimensional maps and their application to hyperspaces, Israel J. Math. 105 (1998) 257-262], Levin proved that in the space C(X, I) of all maps of an n-dimensional compactum X to the unit interval I = [0,1], almost all maps are (n - 1)-dimensional Lelek maps. Moreover, he showed that in the space C(X, I~k) of all maps of an n-dimensional compactum X to the k-dimensional cube I~k (k ≥ 1), almost all maps are (n - k)-dimensional Lelek maps. In this paper, we generalize Levin's result. For any (separable) metric space Y, we define the piecewise embedding dimension ped(Y) of Y and we prove that in the space C(X, Y) of all maps of an n-dimensional compactum X to a complete metric ANR Y, almost all maps are (n - k)-dimensional Lelek maps, where k = ped(Y). As a corollary, we prove that in the space C(X, Y) of all maps of an n-dimensional compactum X to a Peano curve Y, almost all maps are (n - 1)-dimensional Lelek maps and in the space C(X, M) of all maps of an n-dimensional compactum X to a k-dimensional Menger manifold M, almost all maps are (n - k)-dimensional Lelek maps. It is known that k-dimensional Lelek maps are k-dimensional maps for k ≥ 0.