On a conjecture of Bredon

Using the localization technique for equivariant cohomology theory we prove a conjecture of G.Bredon (s. [4], p.381) which states that under certain conditions (s. the theorem below) the cohomology with Z_p-coefficients of each component of the fixpoint set of a Z_p-space can be generated as an algebra by (at most) the same number of elements as the cohomology of the space itself.