TITLE: Equivariant singular cohomology
QUESTION [10 upvotes]: One can define the $G$-equivariant cohomology of a space $X$ as being the ordinary singular cohomology of $X \times_G EG$ --- I think this is due to Borel? (See e.g. section 2 of these notes)
Alternatively if $X$ is a manifold, we also have $G$-equivariant de Rham cohomology, defined in terms of $G$-equivariant differential forms --- I think this is due to Cartan? (See e.g. section 3 of loc. cit.)
I suspect this is extremely standard or obvious, but if it is, I don't know where it's written down: Is it possible to define equivariant cohomology of a topological space in terms of some notion of "equivariant singular cochains", that is, without using the Borel construction?

REPLY [10 votes]: 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 $A$, $H^*(EG\times_G X;A)$
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 $EG\times X$.  That is,
$$ H^*_G(EG\times X; \underline{A}) = H^*(EG\times_G X;A) $$