Separation of homogeneous connected locally compact spaces

We prove that any region $\mathrm{\Gamma }$ in a homogeneous $n$-dimensional and locally compact separable metric space $X$, where $n\ge 2$, cannot be irreducibly separated by a closed $(n-1)$-dimensional subset $C$ with the following property: $C$ is acyclic in dimension $n-1$ and there is a point $b\in C\cap \mathrm{\Gamma }$ having a special local base ${\mathcal{B}}_C^b$ in $C$ such that the boundary of each $U\in {\mathcal{B}}_C^b$ is acyclic in dimension $n-2$. In case $X$ is strongly locally homogeneous, it suffices to have a point $b\in C\cap \mathrm{\Gamma }$ with an ordinary base ${\mathcal{B}}_C^b$ satisfying the above condition. The acyclicity means triviality of the corresponding Čech cohomology groups. This implies all known results concerning the separation of regions in homogeneous connected locally compact spaces.