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Continuous Singular Cohomology

Abstract As different from the ordinary singular cohomology, where a group of coefficients is abelian, in this paper a continuous singular cohomology with coefficients in a topological abelian group is defined. It is proved that for arbitrary topological spaces and any topological abelian group the continuous singular cohomology satisfies all Eilenberg–Steenrod axioms and the additivity axiom, except the excision axiom. However, if a group of coefficients is an absolute retract, then for metric spaces the continuous singular cohomology satisfies the excision axiom and also there is Mayer–Vietoris sequence and the suspension axiom is satisfied.