# The Banach-Mazur Game

A topological space $X$ is said to be a Baire space if for every countable family $\left\{U_0,U_1,U_2,\cdots \right\}$ of open and dense subsets of $X$, the intersection $\bigcap \limits_{n=0}^\infty U_n$ is dense in $X$ (equivalently if every nonempty open subset of $X$ is of second category in $X$). 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 [4]. For more information about the Banach-Mazur game, see [1]. Good references for basic topological terms are [3] and [5]. All topological spaces are assumed to be at least Hausdorff.

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The Banach-Mazur Game

The Banach-Mazur game is a two-person game played on a topological space. Let $X$ be a space. There are two players, $\alpha$ and $\beta$. They take turn choosing nested decreasing nonempty open subsets of $X$ as follows. The player $\beta$ goes first by choosing a nonempty open subset $U_0$ of $X$. The player $\alpha$ then chooses a nonempty open subset $V_0 \subset U_0$. At the nth play where $n \ge 1$, $\beta$ chooses an open set $U_n \subset V_{n-1}$ and $\alpha$ chooses an open set $V_n \subset U_n$. The player $\alpha$ wins if $\bigcap \limits_{n=0}^\infty V_n \ne \varnothing$. Otherwise the player $\beta$ wins.

If the players in the game described above make the moves $U_0,V_0,U_1,V_1,U_2,V_2,\cdots$, 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 $BM(X,\beta)$. In this game, the player $\beta$ makes the first move. If we modify the game by letting $\alpha$ making the first move, we denote this new game by $BM(X,\alpha)$. In either version, the goal of player $\beta$ is to reach an empty intersection of the chosen open sets while player $\alpha$ wants the chosen open sets to have nonempty intersection.

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 $X$ that is of first category in itself, so that $X=\bigcup \limits_{n=0}^\infty X_n$ where each $X_n$ is nowhere dense in $X$. Then player $\beta$ has a winning strategy in the Banach-Mazur game $BM(X,\beta)$. The player $\beta$ always wins the game by making his/her nth play $U_n \subset V_{n-1} - \overline{X_n}$.

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.

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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 $\beta$ in the game $BM(X,\beta)$. A strategy for player $\beta$ is a function $\sigma$ such that $U_0=\sigma(\varnothing)$ (the first move) and for each partial play of the game ($n \ge 1$)

$\displaystyle (*) \ \ \ \ \ \ U_0,V_0,U_1,V_1,\cdots,U_{n-1},V_{n-1}$,

$U_n=\sigma(U_0,V_0,U_1,V_1,\cdots,U_{n-1},V_{n-1})$ is a nonempty open set such that $U_n \subset V_{n-1}$. If player $\beta$ makes all his/her moves using the strategy $\sigma$, then the strategy $\sigma$ for player $\beta$ contains information on all moves of $\beta$. 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, $U_n=\sigma(V_0,V_1,\cdots,V_{n-1})$.

If $\sigma$ is a winning strategy for player $\beta$ in the game $BM(X,\beta)$, then using this strategy will always result in a win for $\beta$. On the other hand, if $\sigma$ is a not a winning strategy for player $\beta$ in the game $BM(X,\beta)$, then there exists a specific play of the Banach-Mazur game

$\displaystyle . \ \ \ \ \ \ U_0,V_0,U_1,V_1,\cdots,U_{n-1},V_{n-1},\cdots$

such that $U_0=\sigma(\varnothing)$, and for each $n \ge 1$, $U_n=\sigma(V_0,\cdots,V_{n-1})$ and player $\alpha$ wins in this play of the game, that is, $\bigcap \limits_{n=0}^\infty V_n \ne \varnothing$.

In the game $BM(X,\alpha)$ (player $\alpha$ making the first move), a strategy for player $\beta$ is a function $\gamma$ such that for each partial play of the game

$\displaystyle (**) \ \ \ \ \ V_0,U_1,V_1,\cdots,U_{n-1},V_{n-1}$,

$U_n=\gamma(V_0,V_1,\cdots,V_{n-1})$ is a nonempty open set such that $U_n \subset V_{n-1}$.

We now present a lemma that helps translate game information into topological information.

Lemma 1
Let $X$ be a space. Let $O \subset X$ be a nonempty open set. Let $\tau$ be the set of all nonempty open subsets of $O$. Let $f: \tau \longrightarrow \tau$ be a function such that for each $V \in \tau$, $f(V) \subset V$. Then there exists a disjoint collection $\mathcal{U}$ consisting of elements of $f(\tau)$ such that $\bigcup \mathcal{U}$ is dense in $O$.

Proof
This is an argument using Zorn’s lemma. If the open set $O$ in the hypothesis has only one point, then the conclusion of the lemma holds. So assume that $O$ has at least two points.

Let $\mathcal{P}$ be the set consisting of all collections $\mathcal{F}$ such that each $\mathcal{F}$ is a disjoint collection consisting of elements of $f(\tau)$. First $\mathcal{P} \ne \varnothing$. To see this, let $V$ and $W$ be two disjoint open sets such that $V \subset O$ and $W \subset O$. This is possible since $O$ has at least two points. Let $\mathcal{F^*}=\left\{ f(V),f(W)\right\}$. Then we have $\mathcal{F^*} \in \mathcal{P}$. Order $\mathcal{P}$ by set inclusion. It is straightforward to show that $(\mathcal{P}, \subset)$ is a partially ordered set.

Let $\mathcal{T} \subset \mathcal{P}$ be a chain (a totally ordered set). We wish to show that $\mathcal{T}$ has an upper bound in $\mathcal{P}$. The candidate for an upper bound is $\bigcup \mathcal{T}$ since it is clear that for each $\mathcal{F} \in \mathcal{T}$, $\mathcal{F} \subset \bigcup \mathcal{T}$. We only need to show $\bigcup \mathcal{T} \in \mathcal{P}$. To this end, we need to show that any two elements of $\bigcup \mathcal{T}$ are disjoint open sets.

Note that elements of $\bigcup \mathcal{T}$ are elements of $f(\tau)$. Let $T_1,T_2 \in \bigcup \mathcal{T}$. Then $T_1 \in \mathcal{F}_1$ and $T_2 \in \mathcal{F}_2$ for some $\mathcal{F}_1 \in \mathcal{T}$ and $\mathcal{F}_2 \in \mathcal{T}$. Since $\mathcal{T}$ is a chain, either $\mathcal{F}_1 \subset \mathcal{F}_2$ or $\mathcal{F}_2 \subset \mathcal{F}_1$. This means that $T_1$ and $T_2$ belong to the same disjoing collection in $\mathcal{T}$. So they are disjoint open sets that are members of $f(\tau)$.

By Zorn’s lemma, $(\mathcal{P}, \subset)$ has a maximal element $\mathcal{U}$, which is a desired disjoint collection of sets in $f(\tau)$. Since $\mathcal{U}$ is maximal with respect to $\subset$, $\bigcup \mathcal{U}$ is dense in $O$. $\blacksquare$

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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 $\beta$. 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.

Theorem 1
The space $X$ is a Baire space if and only if player $\beta$ has no winning strategy in the game $BM(X,\beta)$.

Proof
$\Longleftarrow$ Suppose that $X$ is not a Baire space. We define a winning strategy in the game $BM(X,\beta)$ for player $\beta$. The space $X$ not being a Baire space implies that there is some nonempty open set $U_0 \subset X$ such that $U_0$ is of first category in $X$. Thus $U_0=\bigcup \limits_{n=1}^\infty F_n$ where each $F_n$ is nowhere dense in $X$.

We now define a winning strategy for $\beta$. Let $U_0$ be the first move of $\beta$. For each $n \ge 1$, let player $\beta$ make his/her move by letting $U_n \subset V_{n-1} - \overline{F_n}$ if $V_{n-1}$ is the last move by $\alpha$. It is clear that whenever $\beta$ chooses his/her moves in this way, the intersection of the open sets has to be empty.

$\Longrightarrow$ Suppose that $X$ is a Baire space. Let $\sigma$ be a strategy for the player $\beta$. We show that $\sigma$ cannot be a winning strategy for $\beta$.

Let $U_0=\sigma(\varnothing)$ be the first move for $\beta$. For each open $V_0 \subset U_0$, $\sigma(V_0) \subset V_0$. Apply Lemma 1 to obtain a disjoint collection $\mathcal{U}_0$ consisting of open sets of the form $\sigma(V_0)$ such that $\bigcup \mathcal{U}_0$ is dense in $U_0$.

For each $W=\sigma(V_0) \in \mathcal{U}_0$, we have $\sigma(V_0,V_1) \subset V_1$ for all open $V_1 \subset W$. So the function $\sigma(V_0,\cdot)$ is like the function $f$ in Lemma 1. We can then apply Lemma 1 to obtain a disjoint collection $\mathcal{U}_1(W)$ consisting of open sets of the form $\sigma(V_0,V_1)$ such that $\bigcup \mathcal{U}_1(W)$ is dense in $W$. Then let $\mathcal{U}_1=\bigcup_{W \in \mathcal{U}_0} \mathcal{U}_1(W)$. Based on how $\mathcal{U}_1(W)$ are obtained, it follows that $\bigcup \mathcal{U}_1$ is dense in $U_0$.

Continue the inductive process in the same manner, we can obtain, for each $n \ge 1$, a disjoint collection $\mathcal{U}_n$ consisting of open sets of the form $\sigma(V_0,\dots,V_{n-1})$ (these are moves made by $\beta$ using the strategy $\sigma$) such that $\bigcup \mathcal{U}_n$ is dense in $U_0$.

For each $n$, let $O_n=\bigcup \mathcal{U}_n$. Each $O_n$ is dense open in $U_0$. Since $X$ is a Baire space, every nonempty open subset of $X$ is of second category in $X$ (including $U_0$). Thus $\bigcap \limits_{n=0}^\infty O_n \ne \varnothing$. 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 $\alpha$ wins). We can extract the play of the game because the collection $\mathcal{U}_n$ are disjoint. Thus the strategy $\sigma$ is not a winning strategy for $\beta$. This completes the proof of Theorem 1. $\blacksquare$

Theorem 2
The space $X$ is of second category in itself if and only if player $\beta$ has no winning strategy in the game $BM(X,\alpha)$.

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Some Completeness Properties

Theorem 1 shows that a Baire space is one in which the player $\beta$ has no winning strategy in the Banach-Mazur game (the version in which $\beta$ makes the first move). In such a space, no matter what strategy player $\beta$ wants to use, it can be foiled by player $\alpha$ by producing one specific play in which $\beta$ loses. We now consider spaces in which player $\alpha$ has a winning strategy. A space $X$ is said to be a weakly $\alpha$-favorable if player $\alpha$ has a winning strategy in the game $BM(X,\beta)$. If $\alpha$ always wins, then $\beta$ has no winning strategy. Thus the property of being a weakly $\alpha$-favorable space is stronger than the Baire property.

In any complete metric space, the player $\alpha$ 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 $\alpha$ 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 $\beta$ of the Banach-Mazur game has no winning strategy. Any Baire space that is not weakly $\alpha$-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 $\mathbb{R}$ that is a Bernstein set is such a space. A subset $B$ of the real line is said to be a Bernstein set if $B$ and its complement intersect every uncountable closed subset of the real line. Bernstein sets are discussed here.

Suppose $\theta$ is a strategy for $\alpha$ in the game $BM(X,\beta)$. If at each step, the strategy $\theta$ 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 $U_0,V_0,\cdots,U_{n-1},V_{n-1},U_n$, the strategy $\theta$ can determine the next move for $\alpha$ by only knowing the last move of $\beta$, i.e., $V_n=\theta(U_n)$. A space $X$ is said to be $\alpha$-favorable if player $\alpha$ has a stationary winning strategy in the game $BM(X,\beta)$. Clearly, any $\alpha$-favorable spaces are weakly $\alpha$-favorable spaces. However, there are spaces in which player $\alpha$ has a winning strategy in the Banach-Mazur game and yet has no stationary winning strategy (see [2]). Stationary winning strategy for $\alpha$ is also called $\alpha$-winning tactic (see [1]).

Reference

1. Choquet, G., Lectures on analysis, Vol I, Benjamin, New York and Amsterdam, 1969.
2. Deb, G., Stategies gagnantes dans certains jeux topologiques, Fund. Math. 126 (1985), 93-105.
3. Engelking, R., General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
4. Haworth, R. C., McCoy, R. A., Baire Spaces, Dissertations Math., 141 (1977), 1 – 73.
5. Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.

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Revised 4/4/2014. $\copyright \ 2014 \text{ by Dan Ma}$