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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 Čech cohomology groups. This implies all known results concerning the separation of regions in homogeneous connected locally compact spaces. [ABSTRACT FROM AUTHOR]