Related Open Game in Analytic Determinacy

Question

For this question, please refer to Chapter 33 page 638, Set Theory Millennium Edition, by Thomas Jech.

The proof of analytic games $G_A$ is converted into an open game $G^\ast$ on some suitable space. In the game $G^\ast$, Player I plays $a_0$, then Player II plays a pair $(b_0, h_0)$, then I plays $a_1$, followed by II playing $(b_1, h_1)$, and so on. Each $h_i$ is an order preserving function from $K_i$ into $\kappa$ (refer to the text for the definition of $K_i$). If Player II is able to construct the $h$'s such that $h_{i+1}$ extends $h_i$ for each $i\in\omega$, then Player II wins. This game $G^\ast$, as mentioned, is an open game.

I would like to know specifically what the payoff set is and what the space is (including the topology).

I came up with the following: The space is $\kappa^K$, where $K=\bigcup_{s\subseteq x}K_s$, $x=\langle a_0, b_0, \ldots\rangle$ formed at the end of the run, and the topology is just like that of the Baire space. The payoff set in $G^\ast$ is $A^\ast=\{f\in\kappa^K:f\;\text{is not order preserving}\}$. This set is closed by showing that the complement is open (easy). Hence, the game $G^\ast$ is an open/closed game.

The above seem plausible, but the issue I have with the set I came up with is

1. it doesn't look like a set that include a "pair" as in $(b_0, h_0)$, for instance, does not appear anywhere, and

2. the set $K$ is particular to the $x=\langle a_0, b_0, \ldots\rangle$ produced.

I would appreciate if you could let me know if the above is right and provide me a hint as to how to relate to the related notion of a homogeneous tree thereafter. If the above doesn't makes sense, please let me know where it went wrong and a hint as to how to get the right open set in what topological space would be nice.

Thanks!

Answer 1

One possible space is the space of all sequences $(a_0,(b_0,h_0),a_1,(b_1,h_1),\dots)$, where the $a_i$ and $b_i$ are natural numbers and the $h_i$ are order preserving maps from a finite subset of $Seq$ into $\kappa$. If we identify the order preserving maps from a finite subset of $Seq$ into $\kappa$ with with $\kappa$ (note that there are $\kappa$ such maps), then the space becomes an alternating product of length $\omega$ of the spaces $\omega$ and $\omega\times\kappa$. $\omega\times\kappa$ can be further identified with $\kappa$, so the space can be considered as an alternating product of length $\omega$ of the spaces $\omega$ and $\kappa$.
Here both $\omega$ and $\kappa$ carry the discrete topology.

The pay-off set, i.e., the winning set for I, is the collection of all such sequences that correspond to games that II loses. II wins, i.e., $(a_0,(b_0,h_0),a_1,(b_1,h_1),\dots)$ is not in the payoff set, if the $h_i$ extend each other (getting larger with $i$) and each $h_i$ is an order preserving mapping from $(K_s,\preccurlyeq)$ into $\kappa$ where $s=(a_0,b_0,\dots,a_i,b_i)$ (see Jech's book for the definition of $K_s$).

I am aware of the fact that I don't write much more than Jech does, but I hope that I have filled in enough detail to be understandable.