>[!warning] Disclaimer
>Lecture notes of a course by Miguel Moreno.
## 11.3.25
>[!info] Definition
>A set $I\subseteq\mathcal{P}(S)$ is an *ideal* on $S$ if it satisfies
the following properties:
> - $S\not\in I$, $\varnothing\in I$
> - $X\cup Y\in F$ whenever $X,Y\in I$ (finite union property)
> - $Y\in I$ whenever $Y\subseteq X\subseteq S$ and $X\in I$ (downard closed / closed under subsets)
>
>An ideal is *
lt;\kappa$-complete* if for every $A \subseteq I$ s.t. $|A|<\kappa$, $\bigcup A \in I$
>[!example]
>*The bounded ideal*
>$\mathrm{Bd}(\kappa)=\{x \subseteq \kappa \mid x \text{ is bounded in } \kappa\}$
>[!attention]
>Assume throughout $\kappa^{<\kappa}=\kappa$.
>[!info] Definition - the *Ideal Topology*
>Let $\mathcal{I} \subseteq \mathcal{P}(\kappa)$ an ideal.
>For all $A \in \mathcal{I}$ and $\eta:A \to \kappa$, let
> $
> N_{\eta}:=\{ \xi : \kappa \to \kappa \mid \eta \subseteq \xi\}
> $
>The *ideal topology of $\mathcal{I}$* is the topology on $\kappa^{\kappa}$ generated by the sets $\{ N_{\eta} \mid A\in \mathcal{I}, \eta:A \to \kappa \}$.
>[!example]
>The *Bounded topology* is the ideal topology of $\mathrm{Bd}(\kappa)$.
>**Lemma.** This is the same as the ideal topology of $\kappa$ (viewing $\kappa \subseteq \mathcal{P}(\kappa)$ as an ideal), i.e. generated by $N_{\eta}$ for $\eta:\alpha \to \kappa$ where $\alpha<\kappa$.
>[!info] Definition
> The *generalized Baire space* is $\kappa^{\kappa}$ with the bounded topology.
> The *generalized Cantor space* is $2^{\kappa}$ with the induced topology.
>[!info] Definition
>The class of *$\kappa$-Borel sets* of $\kappa^{\kappa}$ is the smallest set containing all open sets and closed under complementation and unions and intersections of size $\kappa$.
>**The Borel hierarchy**
>$\Sigma^{0}_{1}=\{ X \subseteq \kappa^{\kappa} \mid X \text{ is open}\}$
>$\Pi^{0}_{1}=\{ X \subseteq \kappa^{\kappa} \mid X \text{ is closed}\}$
>$\Sigma^{0}_{\alpha} = \{ \bigcup_{\gamma<\kappa}A_{\gamma} \mid A_{\gamma} \in \bigcup_{\beta<\alpha} \Pi^{0}_{\beta}\}$
>$\Pi^{0}_{\alpha} = \{ \kappa^{\kappa}\smallsetminus X \mid X \in \Sigma^{0}_{\alpha}\}$
**Fact.** $\kappa$-Borel $= \bigcup_{\alpha<\kappa^{+}}\Sigma^{0}_{\alpha}$
>[!info] Definition
>$X \subseteq \kappa^{\kappa}$ is:
>*$\Sigma^{1}_{1}$ - analytic* - if there is a closed $Y \subseteq \kappa^{\kappa}\times \kappa^{\kappa}$ such that $Pr_{1}(Y)=X$
>*$\Pi^{1}_{1}$ - co-analytic* - if there is a $\Sigma^{1}_{1}$ set $Y$ such that $X=\kappa^{\kappa} \smallsetminus Y$
>*$\Delta^{1}_{1}$* if it is both $\Sigma^{1}_{1}$ and $\Pi^{1}_{1}$.
>[!info] Definition
>Recall [[Trees#Sequential/Streamlined trees|sequential trees]].
>- A *$\kappa^{+},\lambda$ tree* is a $\kappa^{+}$ tree with no chains of length $\lambda$ such that every chain of has a unique supremum.
>- A pair $(T,h)$ is *$\kappa \text{-Borel}^{\ast}$ code* if $T$ is a $\kappa^{+},\kappa$-tree and $h:T\to \{\cup,\cap\} \cup \Sigma^{0}_{1}$ s.t. $h(x) \in \{\cup,\cap\}$ iff $x$ is not a leaf.
>- Let $(T,h)$ be a $\kappa \text{-Borel}^{\ast}$ code, and $\eta \in \kappa^{\kappa}$. Define the game $B^{\ast}(T,h,\eta)$ as follows:
> - The game starts at $\varnothing$.
> - If $h(\varnothing)=\cap$ Player I chooses a successor of $\varnothing$.