标题： 关于连续竞赛中君主的存在性 显示英文标题

标题： On the existence of kings in continuous tournaments

摘要： 经典结果Landau关于有限竞赛图（=有限有向完全图）中存在王的结论被扩展到连续竞赛图，其中玩家集合X是一个紧致Hausdorff空间。 也证明了以下部分逆定理。 设X是一个Tychonoff空间，它要么是零维的，要么是局部连通的，要么是伪紧的，要么是线性有序的。 如果X至少存在一个连续竞赛图，并且X上的每个连续竞赛图都有一个王，那么X必须是紧致的。 我们通过给出单位正方形的一个稠密连通子空间Y的例子，证明了我们定理的完全逆是不可能的，该Y恰好存在两个连续竞赛图，这两个竞赛图都有一个王，但Y甚至不是解析的（更不用说紧致的了）。 显示英文摘要

摘要： 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).