Abstract:
We say that a mapping w between two topological abelian groups G and H is a pseudo-homomorphism if the associated map (x, y) is an element of G x G (bar right arrow) w(x y) -w(x) -w(y) is an element of H is continuous. This notion appears naturally in connection with cross sections (continuous right inverses for quotient mappings): given an algebraically splitting, closed subgroup H of a topological group X such that the projection pi : X -> X/H admits a cross section, one obtains a pseudo-homomorphism of X/H to H, and conversely. We show that H splits as a topological subgroup if and only if the corresponding pseudo-homomorphism can be decomposed as a sum of a homomorphism and a continuous mapping. We also prove that pseudo-homomorphisms between Polish groups satisfy the closed graph theorem. Several examples are given. (c) 2017 Elsevier B.V. All rights reserved.