Choquet game

The Choquet game is a topological game named after Gustave Choquet, who was in 1969 the first to investigate such games.[1] A closely related game is known as the strong Choquet game.

Let ${\displaystyle X}$ be a non-empty topological space. The Choquet game of ${\displaystyle X}$, ${\displaystyle G(X)}$, is defined as follows: Player I chooses ${\displaystyle U_{0}}$, a non-empty open subset of ${\displaystyle X}$, then Player II chooses ${\displaystyle V_{0}}$, a non-empty open subset of ${\displaystyle U_{0}}$, then Player I chooses ${\displaystyle U_{1}}$, a non-empty open subset of ${\displaystyle V_{0}}$, etc. The players continue this process, constructing a sequence ${\displaystyle U_{0}\supseteq V_{0}\supseteq U_{1}\supseteq V_{1}\supseteq U_{2}...}$ If ${\displaystyle \bigcap \limits _{i=0}^{\infty }U_{i}=\emptyset }$ then Player I wins, otherwise Player II wins.

It was proved by John C. Oxtoby that a non-empty topological space ${\displaystyle X}$ is a Baire space if and only if Player I has no winning strategy. A nonempty topological space ${\displaystyle X}$ in which Player II has a winning strategy is called a Choquet space. (Note that it is possible that neither player has a winning strategy.) Thus every Choquet space is Baire. On the other hand, there are Baire spaces (even separable metrizable ones) which are not Choquet spaces, so the converse fails.

The strong Choquet game of ${\displaystyle X}$, ${\displaystyle G^{s}(X)}$, is defined similarly, except that Player I chooses ${\displaystyle (x_{0},U_{0})}$, then Player II chooses ${\displaystyle V_{0}}$, then Player I chooses ${\displaystyle (x_{1},U_{1})}$, etc, such that ${\displaystyle x_{i}\in U_{i},V_{i}}$ for all ${\displaystyle i}$. A topological space ${\displaystyle X}$ in which Player II has a winning strategy for ${\displaystyle G^{s}(X)}$ is called a strong Choquet space. Every strong Choquet space is a Choquet space, although the converse does not hold.

All nonempty complete metric spaces and compact T₂ spaces are strong Choquet. (In the first case, Player II, given ${\displaystyle (x_{i},U_{i})}$, chooses ${\displaystyle V_{i}}$ such that ${\displaystyle \operatorname {diam} (V_{i})<1/i}$ and ${\displaystyle \operatorname {cl} (V_{i})\subseteq V_{i-1}}$. Then the sequence ${\displaystyle \left\{x_{i}\right\}\to x\in V_{i}}$ for all ${\displaystyle i}$.) Any subset of a strong Choquet space which is a ${\displaystyle G_{\delta }}$ set is strong Choquet. Metrizable spaces are completely metrizable if and only if they are strong Choquet.[2][3]

References

1. Choquet, Gustave (1969). Lectures on Analysis: Integration and topological vector spaces. W. A. Benjamin. ISBN 9780805369601.
2. Becker, Howard; Kechris, A. S. (1996). The Descriptive Set Theory of Polish Group Actions. Cambridge University Press. p. 59. ISBN 9780521576055.
3. Kechris, Alexander (2012). Classical Descriptive Set Theory. Springer Science & Business Media. pp. 43–45. ISBN 9781461241904.