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.