Here we prove the existence of sheaf cohomology theory in arbitrary o-minimal structures.

We show that in an o-minimal expansion of an ordered group finite definable extensions of a definable group which is defined in a reduct are already defined in the reduct. A similar result is proved for finite topological extensions of definable groups defined in o-minimal expansions of the ordered set of real numbers.

We prove the existence of aČech cohomology theory in arbitrary o-minimal structures with definable Skolem functions satisfying the Eilenberg-Steenrod axioms. * With partial support from the FCT (Fundação para a Ciência e Tecnologia), program POCTI (Portugal/FEDER-EU).† Supported by a postdoctoral fellowship from CMAF, Universidade de Lisboa. MSC: 03C64; 55N05.