Lelek maps and n-dimensional maps from compacta to polyhedra

Abstract

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 .