On the existence of kings in continuous tournaments

TL;DR

This paper extends Landau's classical finite tournament result to continuous tournaments on compact Hausdorff spaces, explores conditions for the existence of kings, and provides counterexamples in non-compact spaces.

Contribution

It generalizes the existence of kings from finite to continuous tournaments on compact spaces and investigates topological conditions affecting this property.

Findings

Landau's finite tournament result extended to continuous tournaments on compact spaces.

Certain topological spaces guarantee the existence of a king in every continuous tournament.

Counterexamples show non-compact spaces can admit tournaments with kings without being compact.

Abstract

The classical result of Landau on the existence of kings in finite tournaments (=finite directed complete graphs) is extended to continuous tournaments for which the set X of players is a compact Hausdorff space. The following partial converse is proved as well. Let X be a Tychonoff space which is either zero-dimensional or locally connected or pseudocompact or linearly ordered. If X admits at least one continuous tournament and each continuous tournament on X has a king, then X must be compact. We show that a complete reversal of our theorem is impossible, by giving an example of a dense connected subspace Y of the unit square admitting precisely two continuous tournaments both of which have a king, yet Y is not even analytic (much less compact).