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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Topological game ： ウィキペディア英語版 Topological game A topological game is an infinite game of perfect information played between two players on a topological space. Players choose objects with topological properties such as points, open sets, closed sets and open coverings. Time is generally discrete, but the plays may have transfinite lengths, and extensions to continuum time have been put forth. The conditions for a player to win can involve notions like topological closure and convergence. It turns out that some fundamental topological constructions have a natural counterpart in topological games; examples of these are the Baire property, Baire spaces, completeness and convergence properties, separation properties, covering and base properties, continuous images, Suslin sets, and singular spaces. At the same time, some topological properties that arise naturally in topological games can be generalized beyond a game-theoretic context: by virtue of this duality, topological games have been widely used to describe new properties of topological spaces, and to put known properties under a different light. The term ''topological game'' was first introduced by Berge,〔 C. Berge, Topological games with perfect information. Contributions to the theory of games, vol. 3, 165–178. Annals of Mathematics Studies, no. 39. Princeton University Press, Princeton, N. J., 1957.〕〔C. Berge, Théorie des jeux à n personnes, Mém. des Sc. Mat., Gauthier-Villars, Paris 1957.〕〔A. R. Pears, On topological games, Proc. Cambridge Philos. Soc. 61 (1965), 165–171.〕 who defined the basic ideas and formalism in analogy with topological groups. A different meaning for ''topological game'', the concept of “topological properties defined by games”, was introduced in the paper of Rastislav Telgársky,〔 R. Telgársky, On topological properties defined by games, Topics in Topology (Proc. Colloq. Keszthely 1972), Colloq. Math. Soc. János Bolyai, Vol. 8, North-Holland, Amsterdam 1974, 617–624.〕 and later "spaces defined by topological games";〔 R. Telgársky, Spaces defined by topological games, Fund. Math. 88 (1975), 193–223.〕 this approach is based on analogies with matrix games, differential games and statistical games, and defines and studies topological games within topology. After more than 35 years, the term “topological game” became widespread, and appeared in several hundreds of publications. The survey paper of Telgársky〔 R. Telgársky, Topological Games: On the 50th Anniversary of the Banach-Mazur Game, Rocky Mountain J. Math. 17 (1987), 227–276. () (3.2MB PDF)〕 emphasizes the origin of topological games from the Banach–Mazur game. There are two other meanings of topological games, but these are used less frequently. * The term ''topological game'' introduced by Leon Petrosjan〔L. A. Petrosjan, Topological games and their applications to pursuit problems. I. SIAM J. Control 10 (1972), 194–202.〕 in the study of antagonistic pursuit-evasion games. The trajectories in these topological games are continuous in time. * The games of Nash (the Hex games), the Milnor games (Y games), the Shapley games (projective plane games), and Gale's games (Bridg-It games) were called ''topological games'' by David Gale in his invited address (). The number of moves in these games is always finite. The discovery or rediscovery of these topological games goes back to years 1948–49. == Basic setup for a topological game == Many frameworks can be defined for infinite positional games of perfect information. The typical setup is a game between two players, I and II, who alternately pick subsets of a topological space ''X''. In the ''n''th round, player I plays a subset ''I''''n'' of ''X'', and player II responds with a subset ''J''''n''. There is a round for every natural number ''n'', and after all rounds are played, player I wins if the sequence :''I''0, ''J''0, ''I''1, ''J''1,... satisfies some property, and otherwise player II wins. The game is defined by the target property and the allowed moves at each step. For example, in the Banach–Mazur game ''BM''(''X''), the allowed moves are nonempty open subsets of the previous move, and player I wins if $\backslash bigcap\_n\; I\_n\; \backslash neq\; \backslash emptyset$. This typical setup can be modified in various ways. For example, instead of being a subset of ''X'', each move might consist of a pair $(I,\; p)$ where $I\; \backslash subset\; X$ and $p\; \backslash in\; x$. Alternatively, the sequence of moves might have length some ordinal number other than $\backslash omega$. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Topological game」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.