# Banach–Mazur game

In general topology, set theory and game theory, a BanachMazur 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 spaces. This game was the first infinite positional game of perfect information to be studied. It was introduced by Mazur as problem 43 in the Scottish book, and Mazur's questions about it were answered by Banach.

## Definition and properties

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 ${\displaystyle Y}$, a fixed subset ${\displaystyle X\subset Y}$, and a family ${\displaystyle W}$ of subsets of ${\displaystyle Y}$ that satisfy the following properties.

The following properties hold.

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 ${\displaystyle MB(X,J)}$, consists in letting ${\displaystyle Y=J}$, i.e. the unit interval ${\displaystyle [0,1]}$, and in letting ${\displaystyle W}$ consist of all closed intervals ${\displaystyle [a,b]}$ contained in ${\displaystyle [0,1]}$. The players choose alternatively subintervals ${\displaystyle J_{0},J_{1},\cdots }$ of ${\displaystyle J}$ such that ${\displaystyle J_{0}\supset J_{1}\supset \cdots }$, and ${\displaystyle P_{1}}$ wins if and only if ${\displaystyle X\cap (\cap _{n<\omega }J_{n})\neq \emptyset }$. ${\displaystyle P_{2}}$ wins if and only if ${\displaystyle X\cap (\cap _{n<\omega }J_{n})=\emptyset }$.

## A simple proof: winning strategies

It is natural to ask for what sets ${\displaystyle X}$ does ${\displaystyle P_{2}}$ have a winning strategy. Clearly, if ${\displaystyle X}$ is empty, ${\displaystyle P_{2}}$ has a winning strategy, therefore the question can be informally rephrased as how "small" (respectively, "big") does ${\displaystyle X}$ (respectively, the complement of ${\displaystyle X}$ in ${\displaystyle Y}$) have to be to ensure that ${\displaystyle P_{2}}$ 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.

Fact: ${\displaystyle P_{2}}$ has a winning strategy if ${\displaystyle X}$ is countable, ${\displaystyle Y}$ is T1, and ${\displaystyle Y}$ has no isolated points.

Proof: Let the elements of ${\displaystyle X}$ be ${\displaystyle x_{1},x_{2},\cdots }$. Suppose that ${\displaystyle W_{1}}$ has been chosen by ${\displaystyle P_{1}}$, and let ${\displaystyle U_{1}}$ be the (non-empty) interior of ${\displaystyle W_{1}}$. Then ${\displaystyle U_{1}\setminus \{x_{1}\}}$ is a non-empty open set in ${\displaystyle Y}$, so ${\displaystyle P_{2}}$ can choose a member ${\displaystyle W_{2}}$ of ${\displaystyle W}$ contained in this set. Then ${\displaystyle P_{1}}$ chooses a subset ${\displaystyle W_{3}}$ of ${\displaystyle W_{2}}$ and, in a similar fashion, ${\displaystyle P_{2}}$ can choose a member ${\displaystyle W_{4}\subset W_{3}}$ that excludes ${\displaystyle x_{2}}$. Continuing in this way, each point ${\displaystyle x_{n}}$ will be excluded by the set ${\displaystyle W_{2n}}$, so that the intersection of all the ${\displaystyle W_{n}}$ will have empty intersection with ${\displaystyle X}$. Q.E.D

The assumptions on ${\displaystyle Y}$ are key to the proof: for instance, if ${\displaystyle Y=\{a,b,c\}}$ is equipped with the discrete topology and ${\displaystyle W}$ consists of all non-empty subsets of ${\displaystyle Y}$, then ${\displaystyle P_{2}}$ has no winning strategy if ${\displaystyle X=\{a\}}$ (as a matter of fact, her opponent has a winning strategy). Similar effects happen if ${\displaystyle Y}$ is equipped with indiscrete topology and ${\displaystyle W=\{Y\}}$.

A stronger result relates ${\displaystyle X}$ to first-order sets.

Fact: Let ${\displaystyle Y}$ be a topological space, let ${\displaystyle W}$ be a family of subsets of ${\displaystyle Y}$ satisfying the two properties above, and let ${\displaystyle X}$ be any subset of ${\displaystyle Y}$. ${\displaystyle P_{2}}$ has a winning strategy if and only if ${\displaystyle X}$ is meagre.

This does not imply that ${\displaystyle P_{1}}$ has a winning strategy if ${\displaystyle X}$ is not meagre. In fact, ${\displaystyle P_{1}}$ has a winning strategy if and only if there is some ${\displaystyle W_{i}\in W}$ such that ${\displaystyle X\cap W_{i}}$ is a comeagre subset of ${\displaystyle W_{i}}$. It may be the case that neither player has a winning strategy: when ${\displaystyle Y}$ is ${\displaystyle [0,1]}$ and ${\displaystyle W}$ consists of the closed intervals ${\displaystyle [a,b]}$, 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 ${\displaystyle [0,1]}$ for which the Banach–Mazur game is not determined.

## References

• [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
• [1987] Telgársky, R. J. Topological Games: On the 50th Anniversary of the Banach–Mazur Game, Rocky Mountain J. Math. 17 (1987), pp. 227–276.[1] (3.19 MB)
• [2003] Julian P. Revalski The Banach–Mazur game: History and recent developments, Seminar notes, Pointe-a-Pitre, Guadeloupe, France, 2003–2004 [2]