I'm trying to understand the proof of Corollary 1.3 part b. in a paper by Bestvina and Mess titled 'The Boundary of negatively curved groups'. I do not understand why $\smash{\check{H}}^{q}(X,A;R)\cong H_{c}^{q}(X-A;R)$. Here $(X, A)$ satisfies the property that $A$ is a $Z$-set in $X$. In particular, I would like to understand why one needs to take Čech cohomology on the left.

I'm trying to understand the proof of Corollary 1.3 part b. in a paper by Bestvina and Mess titled 'The Boundary of negatively curved groups'. I do not understand why $\smash{\check{H}}^{q}(X,A;R)\cong H_{c}^{q}(X-A;R)$. Here $(X, A)$ satisfies the property that $A$ is a $Z$-set in $X$. In particular, I would like to understand why one needs to take Čech cohomology on the left.

I'm trying to understand the proof of Corollary 1.3 part b. in a paper by Bestvina and Mess titled 'The Boundary of negatively curved groups' (https://projecteuclid.org/journals/michigan-mathematical-journal/volume-43/issue-1/Local-homology-properties-of-boundaries-of-groups/10.1307/mmj/1029005393.full). I do not understand why $\check{H}^{q}(X,A;R)\cong H_{c}^{q}(X-A;R)$. Here $(X, A)$ satisfies the property that $A$ is a $Z$-set in $X$. In particular, I would like to understand why one needs to take Cech cohomology on the left.

I'm trying to understand the proof of Corollary 1.3 part b. in a paper by Bestvina and Mess titled 'The Boundary of negatively curved groups'. I do not understand why $\smash{\check{H}}^{q}(X,A;R)\cong H_{c}^{q}(X-A;R)$. Here $(X, A)$ satisfies the property that $A$ is a $Z$-set in $X$. In particular, I would like to understand why one needs to take Čech cohomology on the left.

I'm trying to understand the proof of Corollary 1.3 part b. in a paper by Bestvina and Mess titled 'The Boundary of negatively curved groups' (https://projecteuclid.org/journals/michigan-mathematical-journal/volume-43/issue-1/Local-homology-properties-of-boundaries-of-groups/10.1307/mmj/1029005393.full). I do not understand why $\check{H}^{q}(X,A;R)\cong H_{c}^{q}(X-A;R)$. Here $(X, A)$ satisfies the property that $A$ is a $Z$-set in $X$. In particular, I would like to understand why one needs to take Cech cohomology on the right.

I'm trying to understand the proof of Corollary 1.3 part b. in a paper by Bestvina and Mess titled 'The Boundary of negatively curved groups' (https://projecteuclid.org/journals/michigan-mathematical-journal/volume-43/issue-1/Local-homology-properties-of-boundaries-of-groups/10.1307/mmj/1029005393.full). I do not understand why $\check{H}^{q}(X,A;R)\cong H_{c}^{q}(X-A;R)$. Here $(X, A)$ satisfies the property that $A$ is a $Z$-set in $X$. In particular, I would like to understand why one needs to take Cech cohomology on the left.