#paper #square #forcing # Abstract We show that for any infinite cardinal κ, every strongly (κ+ 1)-strategically closed poset is strongly κ⁺-strategically closed if and only if AP_κ (the approachability property) holds, answering the question asked in [[Ishiu, Yoshinobu - Directive trees and games on posets]]. We also give a complete classification of strengths of strategic closure properties and that of strong strategic closure properties respectively. # Approachability ![[Square principles#Approachability#AP Definition]] ### A generalization of AP **Def** For any regular uncountable cardinal $\lambda$, the principle $A(\lambda)$ asserts the existence of a sequence $\mathcal{C}=\left\langle \left< C_{\alpha},\tau_{\alpha} \right>\mid\alpha <\lambda\right\rangle$ such that 1. $\tau_{\alpha}$ is an infinite cardinal lt;\lambda$; 2. $C_{\alpha}\subseteq\lambda$ and $\mathrm{otp}(C_{\alpha})\leq \tau_{\alpha}$; And there is a club subset $C$ of $\mathrm{acc}(\lambda)$ such that for every $\alpha \in C$, 3. $C_{\alpha}$ is a club subset of $\alpha$; 4. $\forall\beta<\alpha \exists \gamma<\alpha$ such that $C_{\alpha}\cap\beta=C_{\gamma}$ and $\tau_{\alpha}=\tau_{\gamma}$. Also can add without strengthening (lemma 2.9): 5. $\mathrm{otp}(C_{\alpha})=\mathrm{cf}(\alpha)$. Note: $\mathrm{AP_{\kappa}} \iff A(\kappa^{+})$ ## Theorems **Prop 2.6** $B(\kappa)\implies A(\kappa)$ # Generalized directive trees **Definition 3.1.** Let $\kappa \leq \lambda$ be limit ordinals. A tree $T = (λ, ≺)$ is called *$(\lambda,<\kappa)$-directive* if the following conditions hold: 1. $∀α, β < λ (α ≺ β ⇒ α < β)$. 2. For every limit ordinal $η ≤ λ$ such that $\mathrm{cf}(η) < κ$, there is a branch $b$ of $T$ (of limit length) such that $\sup b = η$. 3. Each component of $T$ is of height $\leq\beta$ for some cardinal $\beta<\kappa$ A $(λ, <κ)$-directive tree T is called *continuous* if every branch of $T$ is continuous as a sequence of ordinals. A $(\lambda,<\lambda)$-directive tree is also called *$\lambda$-directive*. ^3a915a **Lemma 3.2** For any singular limit ordinal $\lambda$ there is a continuous $\lambda$-directive tree. **Lemma 3.3.** Let $\lambda$ be an uncountable cardinal. 1. If there is a $\lambda$-directive tree, then every poset which is strongly $(κ + 1)$-strategically closed poset for every cardinal $\kappa<\lambda$ is strongly $\lambda^{*}$-strategically closed. 2. If there is a *continuous* $\lambda$-directive tree, then every poset which is $(κ + 1)$-strategically closed poset for every cardinal $\kappa<\lambda$ is $\lambda^{*}$-strategically closed. Where $\lambda^{*}:=\begin{cases} \lambda & \lambda \text{ regular} \\ \lambda+1 & \lambda \text{ singular} \end{cases}$ **COROLLARY 3.4.** Let $\kappa$ be any infinite cardinal. Then every (strongly) $(\kappa+1)$ -strategically closed poset is (strongly) lt;\kappa ^{+}$-strategically closed . **COROLLARY 3.5.** Let $\lambda$ be any singular cardinal. Then every (strongly) lt;\lambda$- strategically closed poset is (strongly) $(\lambda+1)$-strategically closed, and in fact (strongly) lt;\lambda ^{+}$-strategically closed. ^6ee8b1 **Theorem 4.1** Let $\lambda$ be a regular uncountable cardinal. Then TFAE 1. $B(\lambda)$; 2. There exists a continuous $\lambda$-directive tree; 3. Every poset which is $(\kappa+1)$-strategically closed for every cardinal $\kappa<\lambda$ is $\lambda$-strategically closed. ## Main theorem **Theorem 5.1** Let $\lambda$ be a regular uncountable cardinal. Then TFAE 1. $A(\lambda)$; 2. There exists a $\lambda$-directive tree; 3. Every poset which is strongly $(\kappa+1)$-strategically closed for every cardinal $\kappa<\lambda$ is strongly $\lambda$-strategically closed. 4. Every poset which is strongly $(\kappa+1)$-strategically closed for every cardinal $\kappa<\lambda$ is $\lambda$-strategically closed. In particular, $\mathrm{AP}_{\kappa}\iff$Every strongly $(\kappa+1)$-strategically closed poset strongly $\kappa^{+}$-strategically closed.