Separation of homogeneous connected locally compact spaces

We prove that any region Γ<!-- Γ --> Γ in a homogeneous n n -dimensional and locally compact separable metric space X X , where n ≥<!-- ≥ --> 2 n≥ 2 , cannot be irreducibly separated by a closed ( n −<!-- − --> 1 ) (n-1) -dimensional subset C C with the following property: C C is acyclic in dimension n −<!-- − --> 1 n-1 and there is a point b ∈<!-- ∈ --> C ∩<!-- ∩ --> Γ<!-- Γ --> b∈ C∩ Γ having a special local base B C b B_C^b in C C such that the boundary of each U ∈<!-- ∈ --> B C b U∈ B_C^b is acyclic in dimension n −<!-- − --> 2 n-2 . In case X X is strongly locally homogeneous, it suffices to have a point b ∈<!-- ∈ --> C ∩<!-- ∩ --> Γ<!-- Γ --> b∈ C∩ Γ with an ordinary base B C b 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.