of Čech versions of homology (particularly its failure to satisfy exactness and disjoint union axiom in general), we will make the following assumption on the space $X$.

\[conditionCMP\]\[Condition CMP\] The $G$-space $X$ is said to satisfy condition CMP if $X$ is a Compact, Metrizable Proper $G$-absolute neighbourhood retract (ANR).

These conditions are certainly met in the case of smooth, proper and compact $G$-manifolds with isometric actions, as well as spaces with a proper $G$-CW structure with a finite number of cells.

The input for the spectral sequence is given in terms of Bredon homology with local coefficients associated to a contractible cover. We will introduce some notation relevant to this.

A detailed description of the homological algebra involved in Bredon cohomology, as well as computations related to twisted equivariant $K$-theory and Bredon cohomology can be found in [@barcenasvelasquez], [@barcenasespinozauribevelasquez].

Let ${{\mathcal U}}= \{ U_\sigma \mid  \sigma \in  I \}$ be an open cover of the proper $G$-space $X$ satisfying hypothesis \[conditionCMP\]. Assume that each open set $U_\sigma$ is $G$-equivariantly homotopic to an orbit $G/H_\sigma\subset U_\sigma$ for a finite subgroup $H_\sigma$, and each finite intersection is either empty or $G$-equivariantly homotopic to an orbit.

The existence of such a cover, sometimes 