2023/03/29 by Vesko Valov, Valov, Vesko
Mathematics · #Advanced Operator Algebra Research #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Primary 55M10 #Secondary 54F45
paper · pdf · doi:10.48550/arxiv.2303.16387
openalex publication_date 2023/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that any region Γ in a homogeneous n-dimensional and locally compact separable metric space X, where n≥ 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∈ C∩Γ having a special local base \mathcal BCb in C such that the boundary of each U∈\mathcal BCb is acyclic in dimension n-2. In case X is strongly locally homogeneous, it suffices to have a point b∈ C∩Γ with an ordinary base \mathcal BCb 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.