Aug, 2012
On the existence of kings in continuous tournaments
TOPOLOGY AND ITS APPLICATIONS
- ,
- Volume
- 159
- Number
- 13
- First page
- 3089
- Last page
- 3096
- Language
- English
- Publishing type
- Research paper (scientific journal)
- DOI
- 10.1016/j.topol.2012.05.021
- Publisher
- ELSEVIER SCIENCE BV
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). (C) 2012 Elsevier B.V. All rights reserved.
- Link information
- ID information
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- DOI : 10.1016/j.topol.2012.05.021
- ISSN : 0166-8641
- eISSN : 1879-3207
- Web of Science ID : WOS:000307147000019