Banach-Mazur Games

Origin. Banach and Mazur (1930s), Oxtoby (1957). Infinite games on topological spaces. Players choose nested sets. Winning = nonempty intersection. Foundation for descriptive set theory.

Models. Two players: I and II. Play decreasing sequence of sets. Player II wins if intersection meets target. Characterizes Baire category.

Formalism.

Game G(X, A): X: topological space. A ⊆ X: target set. Players alternate choosing open sets.

Play: Round n: player I chooses Uₙ, player II chooses Vₙ. Constraint: U₁ ⊇ V₁ ⊇ U₂ ⊇ V₂ ⊇ ... All Uₙ, Vₙ open and nonempty.

Winning: Player II wins iff ∩ₙ Vₙ ∩ A ≠ ∅. (Intersection meets target A.)

Banach-Mazur theorem: A is comeager (residual) iff player II has a winning strategy. Player I has a winning strategy iff A is meager in some nonempty open set — not iff A is meager outright. The two conditions come apart: let A ⊆ ℝ be comeager in (0, ∞) and empty on (−∞, 0). Then A is meager in the open set (−∞, 0), so player I wins simply by opening there, yet A is far from meager in ℝ. Player I needs only one region in which to work; player II must survive everywhere.

Determined: One player has winning strategy. Borel games are determined.

Strategies: σ: (U₁,...,Uₙ) → Vₙ (for player II) Winning: guarantees win against all opponent plays.

Variants: Strong Choquet game. Gruenhage game. Point-picking games.

Symbols.

Symbol	Unicode	Name	Meaning
∩	U+2229	Intersection	Of the nested sets, and with the target A
⊇	U+2287	Superset of	The nesting condition on a play
⊆	U+2286	Subset	A ⊆ X: the target lies in the space
∅	U+2205	Empty set	Non-emptiness of the intersection is II's win
≠	U+2260	Not equal to	In the winning condition
σ	U+03C3	Sigma	A strategy, as a function on partial plays
→	U+2192	Function arrow	The type of a strategy
ℝ	U+211D	Double-struck R	The reals, in the counterexample to the meagerness reading
G(X,A)	—	Game	The Banach–Mazur game on X with target A
X	—	Space	A topological space, usually Polish
A	—	Target set	The set player II is trying to hit
U, V	—	Open sets	The moves of player I and player II
n	—	Round index	The stage of play

Metatheory. Mazur posed the game as problem 43 of the Scottish Book and Banach solved it in 1935; Oxtoby (1957) gave the general form. The theorem is an exact category characterization: player II has a winning strategy in the game with target A precisely when A is comeager, and player I has one precisely when A is meager in some nonempty open subset — an asymmetry, since I needs only one region to work in while II must survive everywhere. Determinacy and regularity are linked in both directions: if every set in a class admits a winning strategy for one player in the corresponding unfolded game, every set in that class has the Baire property, and under the Axiom of Determinacy every set of reals does. The related Choquet games characterize completeness rather than category — X is a Baire space exactly when the first player has no winning strategy in the Choquet game, and by Choquet's theorem a nonempty separable metrizable space is Polish exactly when it is strong Choquet.

Applies to. Baire category theory. Descriptive set theory. Topological dynamics. Analysis. Game-theoretic topology.

Limitations. Determinacy fails outright in ZFC: choice produces a Bernstein set that is neither comeager nor meager in any nonempty open set, so neither player wins and the characterization is vacuous for it. The game sees category and nothing else. A comeager set can have measure zero, so no version of the Banach–Mazur game detects measure, and the Lebesgue-measurable analogue requires the separate covering game of Mycielski and Świerczkowski; every regularity property needs its own game, and the perfect set property needs another again. Strategies here are infinite objects over an uncountable space, so the theorem is a fact about existence rather than a method — it yields no algorithm and no complexity bound, and the standard proofs of the equivalences use dependent choice.