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| In [[general topology]], [[set theory]] and [[game theory]], a '''[[Stefan Banach|Banach]]–[[Stanislaw Mazur|Mazur]] game''' is a [[topological game]] played by two players, trying to pin down elements in a set (space). The concept of a Banach–Mazur game is closely related to the concept of [[Baire space]]s. This game was the first infinite [[positional game]] of [[perfect information]] to be studied.
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| == Definition and properties ==
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| In what follows we will make use of the formalism defined in [[Topological game]]. A general Banach–Mazur game is defined as follows: we have a [[topological space]] <math>Y</math>, a fixed subset <math>X \subset Y</math>, and a family <math>W</math> of subsets of <math>Y</math> that satisfy the following properties.
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| * Each member of <math>W</math> has non-empty interior.
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| * Each non-empty open subset of <math>Y</math> contains a member of <math>W</math>.
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| We will call this game <math>MB(X,Y,W)</math>. Two players, <math>P_1</math> and <math>P_2</math>, choose alternatively elements <math>W_0</math>, <math>W_1</math>, <math>\cdots</math> of <math>W</math> such that <math>W_0 \supset W_1 \supset \cdots</math>. <math>P_1</math> wins if and only if <math>X \cap (\cap_{n<\omega} W_n) \neq \emptyset</math>.
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| The following properties hold.
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| * <math>P_2 \uparrow MB(X,Y,W)</math> if and only if <matH>X</math> is of the ''first category'' in <math>Y</math> (a set is of the [[Baire space|first category]] or [[meagre set|meagre]] if it is the countable union of nowhere-dense sets).
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| * Assuming that <math>Y</math> is a complete metric space, <math>P_1 \uparrow MS(X,Y,W)</math> if and only if <math>X</math> is [[comeager]] in some nonempty open subset of <math>Y</math>.
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| * If <math>X</math> has the Baire property in <math>Y</math>, then <math>MB(X,Y,W)</math> is determined.
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| * Any winning strategy of <math>P_2</math> can be reduced to a [[topological game#Definitions and notation|stationary winning strategy]].
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| * The siftable and strongly-siftable spaces introduced by Choquet can be defined in terms of stationary strategies in suitable modifications of the game. Let <math>BM(X)</math> denote a modification of <math>MB(X,Y,W)</math> where <math>X=Y</math>, <math>W</math> is the family of all nonempty open sets in <math>X</math>, and <math>P_2</math> wins a play <math>(W_0, W_1, \cdots)</math> if and only if <math>\cap_{n<\omega} W_n \neq \emptyset</math>. Then <math>X</math> is siftable if and only if <math>P_2</math> has a stationary winning strategy in <math>BM(X)</math>.
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| * A [[topological game#Definitions and notation|Markov winning strategy]] for <math>P_2</math> in <math>BM(X)</math> can be reduced to a stationary winning strategy. Furthermore, if <math>P_2</math> has a winning strategy in <math>BM(X)</math>, then she has a winning strategy depending only on two preceding moves. It is still an unsettled question whether a winning strategy for <math>P_2</math> can be reduced to a winning strategy that depends only on the last two moves of <math>P_1</math>.
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| * <math>X</math> is called ''weakly <math>\alpha</math>-favorable'' if <math>P_2</math> has a winning strategy in <math>BM(X)</math>. Then, <math>X</math> is a Baire space if and only if <math>P_1</math> has no winning strategy in <math>BM(X)</math>. It follows that each weakly <math>\alpha</math>-favorable space is a Baire space.
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| Many other modifications and specializations of the basic game have been proposed: for a thorough account of these, refer to [1987]. The most common special case, called <math>MB(X,J)</math>, consists in letting <math>Y = J</math>, i.e. the unit interval <math>[0,1]</math>, and in letting <math>W</math> consist of all closed intervals <math>[a,b]</math> contained in <math>[0,1]</math>. The players choose alternatively ''subintervals'' <math>J_0, J_1, \cdots</math> of <math>J</math> such that <math>J_0 \supset J_1 \supset \cdots</math>, and <math>P_1</math> wins if and only if <math>X \cap (\cap_{n<\omega} J_n) \neq \emptyset</math>. <math>P_2</math> wins if and only if <math>X\cap (\cap_{n<\omega} J_n) = \emptyset</math>.
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| == A simple proof: winning strategies ==
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| It is natural to ask for what sets <math>X</math> does <math>P_2</math> have a [[Determinacy#Basic notions|winning strategy]]. Clearly, if <math>X</math> is empty, <math>P_2</math> has a winning strategy, therefore the question can be informally rephrased as how "small" (respectively, "big") does <math>X</math> (respectively, the complement of <math>X</math> in <math>Y</math>) have to be to ensure that <math>P_2</math> has a winning strategy. To give a flavor of how the proofs used to derive the properties in the previous section work, let us show the following fact.
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| '''Fact''': ''<math>P_2</math> has a winning strategy if <math>X</math> is countable, <math>Y</math> is [[T1 space|T<sub>1</sub>]], and <math>Y</math> has no [[Isolated point|isolated]] points.''
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| '''Proof''': Let the elements of <math>X</math> be <math>x_1, x_2, \cdots</math>. Suppose that <math>W_1</math> has been chosen by <math>P_1</math>, and let <math>U_1</math> be the (non-empty) interior of <math>W_1</math>. Then <math>U_1 \setminus \{x_1\}</math> is a non-empty open set in <math>Y</math>, so <math>P_2</math> can choose a member <math>W_2</math> of <math>W</math> contained in this set. Then <math>P_1</math> chooses a subset <math>W_3</math> of <math>W_2</math> and, in a similar fashion, <math>P_2</math> can choose a member <math>W_4 \subset W_3</math> that excludes <math>x_2</math>. Continuing in this way, each point <math>x_n</math> will be excluded by the set <math>W_{2n}</math>, so that the intersection of all the <math>W_n</math> will have empty intersection with <math>X</math>. '''Q.E.D'''
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| The assumptions on <math>Y</math> are key to the proof: for instance, if <math>Y=\{a,b,c\}</math> is equipped with [[Discrete space|the discrete topology]] and <math>W</math> consists of all non-empty subsets of <math>Y</math>, then <math>P_2</math> has no winning strategy if <math>X=\{a\}</math> (as a matter of fact, her opponent has a winning strategy). Similar effects happen if <math>Y</math> is equipped with [[Trivial topology|indiscrete]] topology and <math>W=\{Y\}</math>.
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| A stronger result relates <math>X</math> to first-order sets.
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| '''Fact''': Let <math>Y</math> be a topological space, let <math>W</math> be a family of subsets of <math>Y</math> satisfying the two properties above, and let <math>X</math> be any subset of <math>Y</math>. <math>P_2</math> has a winning strategy if and only if <math>X</math> is [[meagre set|meagre]].
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| This does not imply that <math>P_1</math> has a winning strategy if <math>X</math> is not meagre. In fact, <math>P_1</math> has a winning strategy if and only if there is some <math>W_i \in W</math> such that <math>X \cap W_i</math> is a comeagre subset of <math>W_i</math>. It may be the case that neither player has a winning strategy: when <math>Y</math> is <math>[0,1]</math> and <math>W</math> consists of the closed intervals <math>[a,b]</math>, the game is determined if the target set has the [[property of Baire]], i.e. if it differs from an open set by a meagre set (but the converse is not true). Assuming the [[axiom of choice]], there are subsets of <math>[0,1]</math> for which the Banach–Mazur game is not determined.
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| ==References==
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| [1957] Oxtoby, J.C. ''The Banach–Mazur game and Banach category theorem'', Contribution to the Theory of Games, Volume III, Annals of Mathematical Studies '''39''' (1957), Princeton, 159–163
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| [1987] Telgársky, R. J. ''Topological Games: On the 50th Anniversary of the Banach–Mazur Game'', Rocky Mountain J. Math. '''17''' (1987), pp. 227–276.[http://www.telgarsky.com/1987-RMJM-Telgarsky-Topological-Games.pdf] (3.19 MB)
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| [2003] Julian P. Revalski ''The Banach-Mazur game: History and recent developments'', Seminar notes, Pointe-a-Pitre, Guadeloupe, France, 2003-2004 [http://www1.univ-ag.fr/aoc/activite/revalski/Banach-Mazur_Game.pdf]
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| {{DEFAULTSORT:Banach-Mazur game}}
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| [[Category:Topological games]]
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| [[Category:General topology]]
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| [[Category:Descriptive set theory]]
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| [[Category:Determinacy]]
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