A topological space is said to be a Baire space if for every countable family of open and dense subsets of , the intersection is dense in (equivalently if every nonempty open subset of is of second category in ). By the Baire category theorem, every complete metric space is a Baire space. The Baire property (i.e. being a Baire space) can be characterized using the Banach-Mazur game, which is the focus of this post.
Baire category theorem and Baire spaces are discussed in this previous post. We define the Banach-Mazur game and show how this game is related to the Baire property. We also define some completeness properties stronger than the Baire property using this game. For a survey on Baire spaces, see . For more information about the Banach-Mazur game, see . Good references for basic topological terms are  and . All topological spaces are assumed to be at least Hausdorff.
The Banach-Mazur Game
The Banach-Mazur game is a two-person game played on a topological space. Let be a space. There are two players, and . They take turn choosing nested decreasing nonempty open subsets of as follows. The player goes first by choosing a nonempty open subset of . The player then chooses a nonempty open subset . At the nth play where , chooses an open set and chooses an open set . The player wins if . Otherwise the player wins.
If the players in the game described above make the moves , then this sequence of open sets is said to be a play of the game.
The Banach-Mazur game, as described above, is denoted by . In this game, the player makes the first move. If we modify the game by letting making the first move, we denote this new game by . In either version, the goal of player is to reach an empty intersection of the chosen open sets while player wants the chosen open sets to have nonempty intersection.
A Remark About Topological Games
Before relating the Banach-Mazur game to Baire spaces, we give a remark about topological games. For any two-person game played on a topological space, we are interested in the following question.
- Can a player, by making his/her moves judiciously, insure that he/she will always win no matter what moves the other player makes?
If the answer to this question is yes, then the player in question is said to have a winning strategy. For an illustration, consider a space that is of first category in itself, so that where each is nowhere dense in . Then player has a winning strategy in the Banach-Mazur game . The player always wins the game by making his/her nth play .
In general, a strategy for a player in a game is a rule that specifies what moves he/she will make in every possible situation. In other words, a strategy for a player is a function whose domain is the set of all partial plays of the game, and this function tells the player what the next move should be. A winning strategy for a player is a strategy such that this player always wins if that player makes his/her moves using this strategy. A strategy for a player in a game is not a winning strategy if of all the plays of the game resulting from using this strategy, there is at least one specific play of the game resulting in a win for the other player.
Strategies in the Banach-Mazur Game
With the above discussion in mind, let us discuss the strategies in the Banach-Mazur game. We show that the strategies in this game code a great amount of information about the topological space in which the game is played.
First we discuss strategies for player in the game . A strategy for player is a function such that (the first move) and for each partial play of the game ()
is a nonempty open set such that . If player makes all his/her moves using the strategy , then the strategy for player contains information on all moves of . We adopt the convention that a strategy for a player in a game depends only on the moves of the other player. Thus for the partial play of the Banach-Mazur game denoted by above, .
If is a winning strategy for player in the game , then using this strategy will always result in a win for . On the other hand, if is a not a winning strategy for player in the game , then there exists a specific play of the Banach-Mazur game
such that , and for each , and player wins in this play of the game, that is, .
In the game (player making the first move), a strategy for player is a function such that for each partial play of the game
is a nonempty open set such that .
We now present a lemma that helps translate game information into topological information.
Let be a space. Let be a nonempty open set. Let be the set of all nonempty open subsets of . Let be a function such that for each , . Then there exists a disjoint collection consisting of elements of such that is dense in .
This is an argument using Zorn’s lemma. If the open set in the hypothesis has only one point, then the conclusion of the lemma holds. So assume that has at least two points.
Let be the set consisting of all collections such that each is a disjoint collection consisting of elements of . First . To see this, let and be two disjoint open sets such that and . This is possible since has at least two points. Let . Then we have . Order by set inclusion. It is straightforward to show that is a partially ordered set.
Let be a chain (a totally ordered set). We wish to show that has an upper bound in . The candidate for an upper bound is since it is clear that for each , . We only need to show . To this end, we need to show that any two elements of are disjoint open sets.
Note that elements of are elements of . Let . Then and for some and . Since is a chain, either or . This means that and belong to the same disjoing collection in . So they are disjoint open sets that are members of .
By Zorn’s lemma, has a maximal element , which is a desired disjoint collection of sets in . Since is maximal with respect to , is dense in .
Characterizing Baire Spaces using the Banach-Mazur Game
Lemma 1 is the linkage between the Baire property and the strategies in the Banach-Mazur game. The thickness in Baire spaces and spaces of second category allow us to extract a losing play in any strategy for player . The proofs for both Theorem 1 and Theorem 2 are very similar (after adjusting for differences in who makes the first move). Thus we only present the proof for Theorem 1.
The space is a Baire space if and only if player has no winning strategy in the game .
Suppose that is not a Baire space. We define a winning strategy in the game for player . The space not being a Baire space implies that there is some nonempty open set such that is of first category in . Thus where each is nowhere dense in .
We now define a winning strategy for . Let be the first move of . For each , let player make his/her move by letting if is the last move by . It is clear that whenever chooses his/her moves in this way, the intersection of the open sets has to be empty.
Suppose that is a Baire space. Let be a strategy for the player . We show that cannot be a winning strategy for .
Let be the first move for . For each open , . Apply Lemma 1 to obtain a disjoint collection consisting of open sets of the form such that is dense in .
For each , we have for all open . So the function is like the function in Lemma 1. We can then apply Lemma 1 to obtain a disjoint collection consisting of open sets of the form such that is dense in . Then let . Based on how are obtained, it follows that is dense in .
Continue the inductive process in the same manner, we can obtain, for each , a disjoint collection consisting of open sets of the form (these are moves made by using the strategy ) such that is dense in .
For each , let . Each is dense open in . Since is a Baire space, every nonempty open subset of is of second category in (including ). Thus . From this nonempty intersection, we can extract a play of the game such that the open sets in this play of the game have one point in common (i.e. player wins). We can extract the play of the game because the collection are disjoint. Thus the strategy is not a winning strategy for . This completes the proof of Theorem 1.
The space is of second category in itself if and only if player has no winning strategy in the game .
Some Completeness Properties
Theorem 1 shows that a Baire space is one in which the player has no winning strategy in the Banach-Mazur game (the version in which makes the first move). In such a space, no matter what strategy player wants to use, it can be foiled by player by producing one specific play in which loses. We now consider spaces in which player has a winning strategy. A space is said to be a weakly -favorable if player has a winning strategy in the game . If always wins, then has no winning strategy. Thus the property of being a weakly -favorable space is stronger than the Baire property.
In any complete metric space, the player always has a winning strategy. The same idea used in proving the Baire category theorem can be used to establish this fact. By playing the game in a complete metric space, player can ensure a win by making sure that the closure of his/her moves have diameters converge to zero (and the closure of his/her moves are subsets of the previous moves).
Based on Theorem 1, any Baire space is a space in which player of the Banach-Mazur game has no winning strategy. Any Baire space that is not weakly -favorable is a space in which both players of the Banach-Mazur game have no winning strategy (i.e. the game is undecidable). Any subset of the real line that is a Bernstein set is such a space. A subset of the real line is said to be a Bernstein set if and its complement intersect every uncountable closed subset of the real line. Bernstein sets are discussed here.
Suppose is a strategy for in the game . If at each step, the strategy can provide a move based only on the other player’s last move, it is said to be a stationary strategy. For example, in the partial play , the strategy can determine the next move for by only knowing the last move of , i.e., . A space is said to be -favorable if player has a stationary winning strategy in the game . Clearly, any -favorable spaces are weakly -favorable spaces. However, there are spaces in which player has a winning strategy in the Banach-Mazur game and yet has no stationary winning strategy (see ). Stationary winning strategy for is also called -winning tactic (see ).
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