On local cross sections in topological abelian groups

We introduce the notion of local pseudo-homomorphism between two topological abelian groups. We prove that it is closely related with the widely studied notions of local cross sections and splitting extensions in the category of topological abelian groups. In the final section we present an example of a non-splitting extension of \(({\mathbb {R}},\tau _\nu )\) by \({\mathbb {R}}\), where \(\tau _\nu \) is a metrizable group topology on \({\mathbb {R}}\) weaker than the usual one. This extension admits a local cross section.