# Compact $G$-ENR's and Euler characteristic computed with Alexander-Spanier cohomology with compact support

Let $$(Z,A)$$ a compact ENR pair, then $$\chi(Z)=\chi_c(Z-A)+\chi(A)$$ where $$\chi_c$$ is the Euler characteristic taken in Alexander-Spanier cohomology with compact support (ENR means euclidean neighborhood retract). According to the book Transformation Groups by Tammo tom Dieck, page 230, proposition 1.12 $$\chi(X^{K}/NK)=\displaystyle\sum_{(H_i)}{\chi_c(X^{K}_{(H_i)}/NK)}$$ where $$X$$ is a compact $$G$$-ENR, with $$G$$ a compact Lie group, $$K$$ is a (closed?) subgroup of $$G$$ and $$H_i$$'s run through a complete set of conjugacy classes of (closed?) subgroups of $$G$$. Each $$X^{K}_{(H_i)}/NK$$ is supposed to be a subtraction $$X_i-Y_i$$ where $$(X_i,Y_i)$$ is a compact ENR pair, but I do not see what that spaces are. Any suggestion?, please. Do $$K$$ and each $$H_i$$ need to be closed?.