Topological Tournaments

Abstract

A directed graph $R^{\circ}$ on a set $X$ is a set of ordered pairs of distinct points called \emph{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 $R \subset X \times X$ as a total, antisymmetric relation, i.e. $R \cup R^{-1} = X \times X$ and $R \cap R^{-1}$ is the diagonal $1_X = \{ (x,x) : x \in X \}$. The set of arcs is $R^{\circ} = R \setminus 1_X = (X \times X) \setminus R^{-1}$. A topological tournament on a compact Hausdorff space $X$ is a tournament $R$ which is a closed subset of $X \times X$. 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.