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Cross sections and pseudo-homomorphisms of topological abelian groups. (English) Zbl 1372.22002
Summary: We say that a mapping $$\omega$$ between two topological abelian groups $$G$$ and $$H$$ is a pseudo-homomorphism if the associated map $$(x, y) \in G \times G \mapsto \omega(x + y) - \omega(x) - \omega(y) \in H$$ 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 $$\pi : X \rightarrow X / H$$ 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.

##### MSC:
 22A05 Structure of general topological groups 54H11 Topological groups (topological aspects) 54C65 Selections in general topology 46A30 Open mapping and closed graph theorems; completeness (including $$B$$-, $$B_r$$-completeness)
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