On the existence of kings in continuous tournaments

Masato Nagao; Dmitri Shakhmatov

doi:10.1016/j.topol.2012.05.021

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arXiv · 1205.0600

On the existence of kings in continuous tournaments

Masato Nagao·Dmitri Shakhmatov

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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).

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Masato Nagao, Dmitri Shakhmatov. 2012-05-03. On the existence of kings in continuous tournaments. https://doi.org/10.1016/j.topol.2012.05.021

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