  We prove that any region $\Gamma$ in a homogeneous $n$-dimensional and
locally compact separable metric space $X$, where $n\geq 2$, 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
$b\in C\cap\Gamma$ 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 $b\in
C\cap\Gamma$ with an ordinary base $\mathcal B_C^b$ satisfying the above
condition. The acyclicity means triviality of the corresponding \v{C}ech
cohomology groups. This implies all known results concerning the separation of
regions in homogeneous connected locally compact spaces.
