Separation of homogeneous connected locally compact spaces

Abstract

We prove that any region Γ \Gamma in a homogeneous n n -dimensional and locally compact separable metric space X X , where n ≥ 2 n\geq 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 b\in C\cap \Gamma having a special local base B C b \mathcal B_C^b in C C such that the boundary of each U ∈ B C b U\in \mathcal 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 b\in C\cap \Gamma with an ordinary base B C b \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.