Certain Mappings or Decompositions Which Are Topologically Projections

Introduction. A general question which is of interest is the following. Suppose that ƒ is a mapping of a compact metric continuum X onto a metric space F. Under what conditions is there an embedding of X and Y in E (Euclidean n-space) or H (Hubert space) so that ƒ is topologically equivalent to a projection onto Y defined by some collection of parallel hyperplanes? Theorem 1 below provides an answer for a very special case of this general question. Although this theorem is actually a corollary of a more general theorem, we feel that its proof provides motivation and understanding for the main theorem.