CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedaudited 2026-09-10

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Unraveling covers give determinacy

Statement

Assume ZFC. If a game covering unravels $A \subseteq [T]$ , then $G (A; T)$ is determined, with the given terminal taboos.

Facts & Assumptions

[F1]

Open and closed Gale–Stewart games are determined proves open-payoff determinacy on taboo trees in ZFC.

[F2]

Winning strategies descend through game coverings sends a source winning strategy to a target winning strategy for the same player.

[A1]

Assume The Axiom of Choice, as required for F1.

Proof

Given: A covering $(S, π, ϕ)$ with $π^{- 1} (A)$ clopen in $[S]$ .

1.1

In particular $π^{- 1} (A)$ is open in the infinite-play subspace of the taboo tree $S$ . These are exactly the payoff and tree hypotheses of F1, whose ZFC hypothesis is licensed by A1. Obtain a winning strategy $σ$ for one of the two players in $G (π^{- 1} (A); S)$ .

givenF1A1

2.1

Apply F2 to this covering, payoff and winning strategy. It gives $ϕ (σ)$ winning for the same player in $G (A; T)$ ; the existence of such a strategy is determinacy. This includes a terminal root or empty branch space, since F1 and F2 both include terminal maximal plays. QED.

F2step 1.1

Depends on

Used by

Borel games are determined Theorem

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Corollary 3 (standard reference, not scraped)
Lemma 1, printed p451 (standard reference, not scraped)