This paper contains the construction of an equivariant Z-graded cohomology theory satisfying all the equivariant Eilenberg-Steenrod axioms, including the dimension axiom. The theory is defined on the category of all G-pairs, G a finite group, and it can have as coefficients any contravariant coefficient system. The construction of this theory is a generalization of the construction of ordinary Alexander-Spanier cohomology, whence the title of the paper.
Properties: 1) A closed subspace of a paracompact G-space is taut. 2) The equivariant Alexander-Spanier cohomology of a paracompact G-space X is isomorphic to the ordinary cohomology of X/G with coefficients in a suitable sheaf. 3) For a locally nice paracompact G-space the equivariant Alexander-Spanier cohomology agrees with S. Illman’s equivariant singular cohomology.