at.algebraic topology - Equivariant singular cohomology

In 1965 or so, Glen Bredon defined ordinary equivariant cohomology, ordinary meaning
that it satisfies the dimension axiom: For each coefficient system $M$ (contravariant
functor from the orbit category of G to the category of Abelian groups), there is a
unique cohomology theory $H^*_G(-;M)$ such that, when restricted to the orbit category,
it spits out the functor $M$. Just as in the nonequivariant world, it can be defined
using either singular or cellular cochains, the latter defined using $G$-CW complexes.
This works as stated for any topological group $G$.

For an abelian group $A$, Borel cohomology with coefficients in $A$, $H^*(EGtimes_G X;A)$
is the extremely special case in which one takes $M$ to be the constant coefficient system
$underline{A}$ at the group $A$ and replaces $X$ by $EGtimes X$. That is,

$$ H^*_G(EGtimes X; underline{A}) = H^*(EGtimes_G X;A) $$