$\mathbf {The \ Problem \ is}:$ Let, $X$ be a compact , connected homogeneous metric ANR (absolute neighbourhood retract) has cohomological dimension $\operatorname{dim}_G =n$ with respect to an abelian group $G$ (i.e. for every closed set $A$ in $X$ and every continuous map $\alpha : A\to K(G,n)$ , there exists a continuous extension $\tilde{\alpha} : X\to K(G,n)$ of $\alpha.$ ) . Now let $Z$ be a closed subset of $X$ with non-empty interior . Show that there exists an open set $U$ in $Z$ such that $\operatorname{dim}_G U=n.$
$\mathbf {My \ approach}:$ Is this true that every open set $U$ of a space $X$ also has $\operatorname{dim}_G U= n$ ? I was trying to prove this using excision for cohomology but failed . I was also trying to use that fact that $K(G,n)$ is an ANR space but couldn't proceed much . I guess this is not true but we need to use the fact that $Z$ is closed and $X$ is an ANR . Thanks in advance for any hints .