2023/03/31 by Ethan Akin, Akin, Ethan
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2304.00055
openalex publication_date 2023/03/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A directed graph R∘ on a set X is a set of ordered pairs of distinct points called arcs. It is a tournament when every pair of distinct points is connected by an arc in one direction or the other (and not both). We can describe a tournament R ⊂ X × X as a total, antisymmetric relation, i.e. R ∪ R-1 = X × X and R ∩ R-1 is the diagonal 1X = \ (x,x) : x ∈ X \. The set of arcs is R∘ = R ∖ 1X = (X × X) ∖ R-1. A topological tournament on a compact Hausdorff space X is a tournament R which is a closed subset of X × X. We construct uncountably many non-isomorphic examples on the Cantor set X as well as examples of arbitrarily large cardinality. We also describe compact Hausdorff spaces which do not admit any topological tournament.