On mappings of compact spaces into Cartesian spaces

Eilenberg proved that if a compact space X admits a zero-dimensional map f :X→Y, where Y is m-dimensional, then there exists a map h :X→Im+1 such that f×h :X→Y×Im+1 is an embedding. In this paper we prove generalizations of this result for σ-compact subsets of arbitrary spaces. An example of a compact space X and of a zero-dimensional σ-compact subset A⊂X is given such that for any continuous function f :X→R which is one-to-one on the set A and any Gδ-subset B of X with B⊃A the restriction f|B :B→R has infinite fibers. This example is used to demonstrate that our results are sharp.