Proof By Theorem 3.5, taking $\Omega :=\omega $ .

Proof As $\lambda ^\theta =\lambda $ , by the Engelking–Karlowicz theorem, [Reference Engelking and Karłowicz9], we may fix a sequence $\vec f=\langle f_\alpha \mathrel {|} \alpha <\lambda \rangle $ of functions from $2^\lambda $ to $\lambda $ such that for every function $g:x\rightarrow \lambda $ with $x\in [2^\lambda ]^\theta $ , there exists an $\alpha <\lambda $ such that $g\subseteq f_\alpha $ . Let us identify a useful feature of $\vec f$ .

Now let I denote the collection of all $A\subseteq \lambda $ for which there exists $\mathcal B\in [[2^\lambda ]^\theta ]^\theta $ such that, for every $\alpha \in A$ , for some $B\in \mathcal B$ it is the case that $f_\alpha [B]\neq \theta $ . It is clear that I is a $\theta ^+$ -complete ideal over $\lambda $ and it is proper by Claim 3.4.1. Now pick $c:\lambda \times 2^\lambda \rightarrow \theta $ such that $c(\alpha ,\beta )=f_\alpha (\beta )$ whenever $f_\alpha (\beta )<\theta $ . Then c and I are as sought.

Proof By the Erdős–Rado theorem, the cardinal $\chi :=(\beth _{n-1}(\theta ))^+$ satisfies $\chi \rightarrow (\mu )^n_\theta $ . Put $\sigma :=2^{<\chi }$ and $\lambda :=2^\sigma $ . Consequently, $\sigma = \beth _n(\theta )$ , $\lambda = \beth _{n+1}(\theta )$ , and $\kappa = \beth _{n+2}(\theta ) = 2^\lambda $ . Altogether,

$$ \begin{align*}\max\{\Omega, \mu\}\le\theta<\chi\le\sigma<\lambda=\lambda^\sigma<\kappa<\Omega^{+\omega}.\end{align*} $$

Fix an $m<\omega $ such that $\kappa =\lambda ^{+m+1}$ and let $\Lambda :=\lambda ^{+m}$ , so that $\lambda \leq \Lambda $ . Note that Hausdorff’s lemma implies that $\Lambda ^\sigma =\Lambda <\kappa $ . In addition, since $\lambda ^\sigma =\lambda $ , Lemma 3.4 provides us with a $\sigma ^+$ -complete proper ideal I over $\lambda $ such that ${{{\sf onto}}}(I^+,[\kappa ]^{<\sigma },\sigma )$ holds.

Proof The proof is similar to that of [Reference Shelah23, Lemma 2]. As $\kappa =\Lambda ^+$ and $\Lambda ^\chi =\Lambda $ , for every $\rho \in \operatorname {\mathrm {acc}}(\kappa )\cap E^\kappa _{\le \chi }$ we may let $\langle C_{\rho ,j}\mathrel {|} j<\Lambda \rangle $ be an enumeration of all clubs in $\rho $ of order-type no more than $\chi $ . In addition, using $\Lambda ^\chi =\Lambda $ , by the Engelking–Karlowicz theorem, we may fix a sequence $\vec f=\langle f_i\mathrel {|} i<\Lambda \rangle $ of functions from $\kappa $ to $\Lambda $ such that for every function $g:x\rightarrow \Lambda $ with $x\in [\kappa ]^\chi $ , there exists an $i<\Lambda $ such that $g\subseteq f_i$ . Denote $C_\rho ^i:=C_{\rho ,f_i(\rho )}$ . Clearly, for every $i<\Lambda $ , $\vec C^i:=\langle C_\rho ^i\mathrel {|} \rho \in \operatorname {\mathrm {acc}}(\kappa )\cap E^\kappa _{\le \chi }\rangle $ is a $\chi $ -bounded C-sequence.

We claim that there exists an $i<\Lambda $ such that for every club $D\subseteq \kappa $ , for every $\varepsilon <\kappa $ , there exists a $\rho \in R\cap E^\kappa _{\chi }$ such that:

1. (1) $\operatorname {\mathrm {nacc}}(C_\rho ^i)\subseteq D\cap T$ ;

2. (2) $\min (C^i_\rho )\ge \varepsilon $

3. (3) for every $\delta \in \operatorname {\mathrm {acc}}(C^i_\rho )$ , $C^i_\delta =C^i_\rho \cap \delta $ .

Indeed, otherwise, for each $i<\Lambda $ , we may fix a club $D_i\subseteq \kappa $ and some $\varepsilon _i<\kappa $ such that for every $\rho \in R\cap E^\kappa _\chi $ , either (1) fails for $D_i$ or (2) fails for $\varepsilon _i$ or (3) fails. Let $D:=\bigcap _{i<\Lambda }D_i$ and $\varepsilon :=\sup _{i<\Lambda }\varepsilon _i$ . As T is stationary, we may now fix some $\rho \in E^\kappa _\chi $ above $\varepsilon $ such that $D\cap T$ is cofinal in $\rho $ . Fix a club C in $\rho $ of order-type $\chi $ such that $\operatorname {\mathrm {nacc}}(C)\subseteq D\cap T$ and $\min (C)=\varepsilon $ . Pick a function $g:\operatorname {\mathrm {acc}}(C)\cup \{\rho \}\rightarrow \Lambda $ such that $g(\delta )=j$ implies $C\cap \delta =C_{\delta ,j}$ . Pick an $i<\Lambda $ such that $g\subseteq f_i$ . This implies that

$$ \begin{align*}C^i_\rho = C_{\rho, f_i(\rho)} = C_{\rho, g(\rho)} = C,\end{align*} $$

and for $\delta \in \operatorname {\mathrm {acc}}(C)$ ,

$$ \begin{align*}C^i_\delta = C_{\delta, f_i(\delta)} = C_{\delta, g(\delta)} = C\cap \delta.\end{align*} $$

Then we arrive at the following contradiction:

1. (i) $\operatorname {\mathrm {nacc}}(C_\rho ^i)= \operatorname {\mathrm {nacc}}(C)\subseteq D\cap T\subseteq D_i\cap T$ ;

2. (ii) $\min (C^i_\rho )=\min (C)=\varepsilon \ge \varepsilon _i$ ;

3. (iii) for every $\delta \in \operatorname {\mathrm {acc}}(C^i_\rho )$ , $C^i_\delta =C\cap \delta =C^i_\rho \cap \delta $ .

Thus, pick $i<\Lambda $ such that for every club $D\subseteq \kappa $ , for every $\varepsilon <\kappa $ , there exists a $\rho \in R\cap E^\kappa _{\chi }$ such that (1)–(3) holds. Set

$$ \begin{align*}R:=\{ \rho\in\operatorname{\mathrm{acc}}(\kappa)\cap E^\kappa_{\le\chi}\mathrel{|} \forall\delta\in\operatorname{\mathrm{acc}}(C^i_\rho)\,[C^i_\rho\cap\delta=C^i_\delta]\}.\end{align*} $$

Then $\langle C^i_\rho \mathrel {|} \rho \in R\rangle $ is as sought.

Let $\langle C_\rho \mathrel {|} \rho \in R\rangle $ be given by the claim. It follows that the set

$$ \begin{align*}\mathcal E:=\{ \varepsilon<\kappa\mathrel{|} \forall D\subseteq\kappa\text{ club }\exists \rho\in R\cap E^\kappa_{\chi}\,[\operatorname{\mathrm{nacc}}(C_\rho)\subseteq^* D\cap T\ \&\ \min(C_\rho)=\varepsilon]\}\end{align*} $$

is cofinal in $\kappa $ . For every $j<\kappa $ , let $\mathcal E(j)$ denote the unique $\varepsilon \in \mathcal E$ to satisfy $\operatorname {\mathrm {otp}}(\mathcal E\cap \varepsilon )=j$ , and then let:

• • $R_j:=\{\rho \in R\cap E^\kappa _\chi \mathrel {|} \min (C_\rho )=\mathcal E(j)\}$ , and

• • $S_j:=\bigcup \{ \operatorname {\mathrm {acc}}(C_\rho )\mathrel {|} \rho \in R_j\}\cap E^\kappa _\mu $ .

The sets in $\langle S_j\mathrel {|} j<\kappa \rangle $ are pairwise disjoint, since for $j < j'< \kappa $ we have that $\mathcal E(j) \neq \mathcal E(j')$ , and by the coherence property of $\langle C_\rho \mathrel {|} \rho \in R\rangle $ . Also, for every ${j< \kappa }$ and every club $D \subseteq \kappa $ there is a $\rho \in R_j \cap E^\kappa _\chi $ such that $\operatorname {\mathrm {nacc}}(C_\rho ) \subseteq ^* D$ , and hence $\operatorname {\mathrm {acc}}(C_\rho ) \cap D\cap E^\kappa _\mu \neq \emptyset $ as $\mu <\chi $ . It follows that $\langle S_j\mathrel {|} j<\kappa \rangle $ is a partition into stationary sets of some subset S of $E^\kappa _\mu $ .

For each $\delta \in S$ , let

$$ \begin{align*}\mathcal P_\delta:=\bigcup\{{}^{[C_\delta\setminus\epsilon]^n}\delta\mathrel{|} \epsilon\in C_\delta\}\end{align*} $$

and note that

$$ \begin{align*}|\mathcal P_\delta|\le\Lambda^{\chi}\le \Lambda^\sigma= \Lambda<\Lambda^+ =\kappa.\end{align*} $$

For each $\delta \in S$ , since $\operatorname {\mathrm {otp}}(C_\delta )\le \chi $ but $\operatorname {\mathrm {cf}}(\delta )=\mu <\chi $ , it is the case that $\operatorname {\mathrm {otp}}(C_\delta )<\chi $ . Combining this with $\theta \cdot 2^{<\chi }=\sigma $ , let us also fix an enumeration $\langle (\tau _{\delta ,i},B_{\delta ,i})\mathrel {|} i<\sigma \rangle $ of all pairs $(\tau ,B)$ such that $\tau <\theta $ and B is a cofinal subset of $\operatorname {\mathrm {nacc}}(C_\delta )$ of order type  $\mu $ .

Moving on, as $\kappa =2^\lambda =\theta ^\lambda $ , let us fix a bijection $\psi :{}^\lambda \theta \leftrightarrow \kappa $ . For every $\alpha <\lambda $ , define a map $\psi _\alpha :\kappa \rightarrow \theta $ via

$$ \begin{align*}\psi_\alpha(\xi):=\psi^{-1}(\xi)(\alpha).\end{align*} $$

For every subset $C\subseteq \kappa $ , for every map $\eta :[C]^n\rightarrow \kappa $ , for every $\alpha <\lambda $ , we let $\eta ^\alpha :=\psi _\alpha \circ \eta $ , so that $\eta ^\alpha $ is a function from $[C]^n$ to $\theta $ . We say that $f:[C]^n\rightarrow \theta $ is good iff there is some cofinal subset $H\subseteq \operatorname {\mathrm {nacc}}(C)$ such that $f \mathbin {\upharpoonright }[H]^n$ is constant.

Let $\delta \in S$ and $\eta \in \mathcal P_\delta $ . So, for some $\epsilon \in C_\delta $ , $\eta : [C_\delta \setminus \epsilon ]^n \rightarrow \delta $ . As $\operatorname {\mathrm {Im}}(\eta )\subseteq \kappa $ , for every $\alpha <\lambda $ , $\eta ^\alpha :[C_\delta \setminus \epsilon ]^n\rightarrow \theta $ , so we let $A_\eta :=\{\alpha <\lambda \mathrel {|} \eta ^\alpha \text { is good}\}$ . Pick a map $h_\eta :\lambda \rightarrow \sigma $ such that for every $\alpha \in A_\eta $ , $i:=h_\eta (\alpha )$ satisfies that $B_{\delta ,i}$ is a cofinal subset of $\operatorname {\mathrm {nacc}}(C_\delta \setminus \epsilon )$ for which $\eta ^\alpha \mathbin {\upharpoonright }[B_{\delta ,i}]^n$ is constant with value $\tau _{\delta ,i}$ .

Let $\langle g_\delta \mathrel {|} \delta \in S\rangle $ be given by the preceding claim.

Proof Suppose not. Then, for every $\alpha <\lambda $ , we may fix a function $f_\alpha :[\kappa ]^n\rightarrow \theta $ and a club $D_\alpha \subseteq \kappa $ such that for every $\delta \in S_j$ , either $B_{\delta ,g_\delta (\alpha )}\nsubseteq D_\alpha \cap T$ or $f_\alpha "[B_{\delta ,g_\delta (\alpha )}]^n\neq \{\tau _{\delta ,g_\delta (\alpha )}\}$ . Using that $\psi :{}^\lambda \theta \leftrightarrow \kappa $ is a bijection define a map $\eta :[\kappa ]^n\rightarrow \kappa $ via:

$$ \begin{align*}\eta(\gamma_1,\ldots,\gamma_n):=\psi(\langle f_\beta(\gamma_1,\ldots,\gamma_n)\mathrel{|} \beta<\lambda\rangle).\end{align*} $$

Consider the club $D:=\{\delta \in \bigcap _{\alpha <\lambda }D_\alpha \mathrel {|} \eta "[\delta ]^n\subseteq \delta \}$ . Pick $\rho \in R_j$ and a large enough $\epsilon \in C_\rho $ such that $\operatorname {\mathrm {nacc}}(C_\rho \setminus \epsilon )\subseteq D\cap T$ . As $\operatorname {\mathrm {otp}}(\operatorname {\mathrm {nacc}}(C_\rho \setminus \epsilon ))=\chi $ , recalling that $\chi \rightarrow (\mu )^n_\theta $ holds, for every $\alpha <\lambda $ , we may pick a subset $H_\alpha \subseteq \operatorname {\mathrm {nacc}}(C_\rho \setminus \epsilon )$ of order-type $\mu $ such that $f_\alpha \mathbin {\upharpoonright }[H_\alpha ]^n$ is constant, and clearly $\delta _\alpha :=\sup (H_\alpha )$ is an element of $\operatorname {\mathrm {acc}}(C_\rho \setminus \epsilon )\cap E^\kappa _\mu $ , and hence of $S_j\cap D$ too. As $|C_\rho |=\chi \le \sigma $ and I is a proper $\sigma ^+$ -complete ideal on $\lambda $ , let us now pick a $\delta \in S_j\cap D$ such that $A^*:=\{\alpha <\lambda \mathrel {|} \delta _\alpha =\delta \}$ is in $I^+$ .

Since $\delta \in \operatorname {\mathrm {acc}}(C_\rho )$ , it is the case that $C_\delta =C_\rho \cap \delta $ , and hence, for every $\alpha \in A^*$ , $H_\alpha $ witnesses that $f_\alpha \mathbin {\upharpoonright }[C_\delta \setminus \epsilon ]^n$ is good.

Since $\delta \in D$ , we know that $\bar \eta :=\eta \mathbin {\upharpoonright }[C_\delta \setminus \epsilon ]^n$ is in $\mathcal P_\delta $ . For every $\alpha <\lambda $ , recall that $\bar \eta ^\alpha $ is defined as $\psi _\alpha \circ \bar \eta $ , so that, for every $(\gamma _1,\ldots ,\gamma _n)\in [C_\delta \setminus \epsilon ]^n$ ,

$$ \begin{align*}\bar\eta^\alpha(\gamma_1,\ldots,\gamma_n)=\psi_\alpha(\psi(\langle f_\beta(\gamma_1,\ldots,\gamma_n)\mathrel{|} \beta<\lambda\rangle))=f_\alpha(\gamma_1,\ldots,\gamma_n).\end{align*} $$

That is, $\bar \eta ^\alpha =f_\alpha \mathbin {\upharpoonright }[C_\delta \setminus \epsilon ]^n$ for every $\alpha <\lambda $ , so $A_{\bar \eta }$ is equal to $\{\alpha <\lambda \mathrel {|} f_\alpha \mathbin {\upharpoonright }[C_\delta \setminus \epsilon ]^n\text { is good}\}$ and it covers the $I^+$ -set $A^*$ , and hence $A_{\bar \eta }\in I^+$ . Recalling that $g_\delta $ was given to us by Claim 3.5.2, we may now pick an $\alpha \in A_{\bar \eta }$ with $h_{\bar \eta }(\alpha )= g_\delta (\alpha )$ . By the definition of $h_{\bar \eta }$ , this means that $B_{\delta ,g_\delta (\alpha )}$ is a cofinal subset of $\operatorname {\mathrm {nacc}}(C_\delta \setminus \epsilon )$ for which $\bar \eta ^\alpha \mathbin {\upharpoonright }[B_{\delta ,g_\delta (\alpha )}]^n$ is constant with value $\tau _{\delta ,g_\delta (\alpha )}$ . But

$$ \begin{align*}B_{\delta, g_\delta(\alpha)} \subseteq\operatorname{\mathrm{nacc}}(C_\delta\setminus\epsilon)\subseteq\operatorname{\mathrm{nacc}}(C_\rho\setminus\epsilon)\subseteq D\cap T\subseteq D_\alpha\cap T\end{align*} $$

and $\bar \eta ^\alpha \mathbin {\upharpoonright } [B_{\delta ,g_\delta (\alpha )}]^n=f_\alpha \mathbin {\upharpoonright } [B_{\delta ,g_\delta (\alpha )}]^n$ , contradicting the choice of $D_\alpha $ and $f_\alpha $ .

For every $j<\kappa $ , let $\alpha _j<\lambda $ be given by the preceding claim, and then for every $\delta \in S_j$ , let $B_\delta :=B_{\delta ,g_\delta (\alpha _j)}$ and $g(\delta ):=\tau _{\delta ,g_\delta (\alpha _j)}$ . Then $\langle B_\delta \mathrel {|} \delta \in S\rangle $ , $\langle S_j\mathrel {|} j<\kappa \rangle $ and $g:E^\kappa _\mu \rightarrow \theta $ are as sought modulo the fact that $S=\bigcup _{j<\kappa }S_j$ may possibly be a proper subset of $E^\kappa _\mu $ , but this can be mitigated by allocating all the left-out points to $S_0$ and defining $B_\delta $ and $g(\delta )$ arbitrarily over these points.

Fact 4.1. For a ladder system $\vec L=\langle A_\delta \mathrel {|} \delta \in S\rangle $ , all of following are equivalent $:$

1. (1) $X_{\vec L}$ is Hausdorff $;$

2. (2) $X_{\vec L}$ is Hausdorff and regular $;$

3. (3) for every pair $\gamma <\delta $ of ordinals from S, $\sup (A_\gamma \cap A_\delta )<\gamma $ .

Fact 4.2. For a ladder system $\vec L=\langle A_\delta \mathrel {|} \delta \in S\rangle $ , all of following are equivalent $:$

1. (1) $X_{\vec L}$ is a $\Delta $ -space $;$

2. (2) $X_{\vec L}$ is countably metacompact $;$

3. (3) For every function $g:S \rightarrow \omega $ , there is a function $f: \kappa \rightarrow \omega $ , such that, for every $\delta \in S$ , $\sup \{\alpha \in A_\delta \mathrel {|} f(\alpha ) \leq g(\delta )\} <\delta $ .

Proof Since $\vec L$ is a $\mu $ -bounded ladder system over $E^\kappa _\mu $ , Fact 4.1 implies that $X_{\vec L}$ is regular and Hausdorff. Write $\vec L$ as $\langle A_\delta \mathrel {|} \delta \in E^\kappa _\mu \rangle $ . Define a function $F:(\bigcup _{\delta \in S}{}^{A_\delta }\omega )\rightarrow \omega $ by letting for all $\delta \in S$ and $ f:A_\delta \rightarrow \omega $ ,

$$ \begin{align*}F( f):=\min\{n<\omega\mathrel{|} \sup\{\alpha\in A_\delta \mathrel{|} f(\alpha) =n\} =\delta\}.\end{align*} $$

Since $\Phi (\vec L,\omega ,\omega )$ holds, we may now fix a function $g:S\rightarrow \omega $ such that, for every function $f:\kappa \rightarrow \omega $ , there are stationarily many $\delta \in S$ such that $F(f \mathbin {\upharpoonright } A_\delta )= g(\delta )$ . In particular, for every function $f:\kappa \rightarrow \omega $ there are stationarily many $\delta \in S$ such that $\sup \{\alpha \in A_\delta \mathrel {|} f(\alpha ) =g(\delta )\} =\delta $ . So, by Fact 4.2, $X_{\vec L}$ is not countably metacompact.

Proof By Corollary 2.12, there are co-boundedly many uncountable $\mu \in \operatorname {\mathrm {Reg}}(\Lambda )$ , for which there exists a $\mu $ -bounded ladder system $\vec L$ over $E^\kappa _\mu $ such that $\diamondsuit (\vec L, \omega )$ holds, in particular, $\Phi (\vec L, \omega ,\omega )$ holds. Now, appeal to Lemma 4.4.

Proof Denote $\mu :=\aleph _1$ and $\lambda :=2^\mu $ . Note that $\aleph _1<\lambda <\operatorname {\mathrm {cf}}(\kappa )\le \kappa <\aleph _{\omega _1}$ and hence $\kappa $ is regular. As $2^\lambda <\mu ^{+\mu }$ , for every cardinal $\varkappa \in [\mu ,2^\lambda )$ , it is the case that $\mu \le \varkappa <\mu ^{+\mu }$ , and so Fact 2.13 implies that $m(\varkappa , \mu ) = \varkappa <2^\lambda $ . As made clear right after Claim 2.9.1, we then get a $\mu $ -bounded C-sequence $\vec C$ over some stationary $S\subseteq E^\kappa _\mu $ such that $\diamondsuit ^*(\vec C, 2^\lambda , 2^\lambda )$ holds. In particular, there is a $\mu $ -bounded C-sequence $\vec C$ over $E^\kappa _\mu $ such that $\diamondsuit (\vec C, 2^\lambda , 2^\lambda )$ holds. As $\lambda ^\mu =\lambda $ , Lemma 2.8 implies that ${{{\sf onto}}}(\{\lambda \},[2^\lambda ]^{\le \lambda },\mu )$ holds. Then, by Lemma 2.10, $\Phi (\vec C,2^\lambda ,\mu )$ holds. Now, appeal to Lemma 4.4.

Proof Suppose that $\kappa :=2^{2^{2^{\aleph _0}}}$ is smaller than $\aleph _\omega $ . Appealing to Theorem 3.5 with $(n,\Omega ,\mu ,\theta ):=(1,\omega ,\omega ,\omega )$ , we obtain an $\omega $ -bounded ladder system $\vec L=\langle A_\delta \mathrel {|} \delta \in E^\kappa _\omega \rangle $ and a map $g:\kappa \rightarrow \omega $ such that for every function $f:\kappa \rightarrow \omega $ , there are stationarily many $\delta \in E^\kappa _\omega $ such that $f" A_\delta =\{g(\delta )\}$ . Since $\vec L$ is a $\mu $ -bounded ladder system over $E^\kappa _\mu $ , Fact 4.1 implies that $X_{\vec L}$ is regular and Hausdorff. In addition, by Fact 4.2, $X_{\vec L}$ is not countably metacompact.

Proof By [Reference Rinot and Shalev21, Theorem 2.29], the existence of a $\kappa $ -Souslin tree $\mathbf T$ implies that $\clubsuit _{\operatorname {\mathrm {AD}}}(\mathcal S,1,1)$ holds for some $\kappa $ -sized pairwise disjoint family $\mathcal S$ of stationary subsets of $\kappa $ . Denote $S:=\biguplus \mathcal S$ . By [Reference Rinot and Shalev21, Corollary 2.25(2)], if $\mathbf T$ is regressive, then we may moreover secure that $S=E^\kappa _\omega $ . As $\mathcal S$ is infinite, let $\langle S_n\mathrel {|} n<\omega \rangle $ be a partition of S in such a way that each $S_n$ covers some set from $\mathcal S$ , so that $\clubsuit _{\operatorname {\mathrm {AD}}}(\{S_n\mathrel {|} n<\omega \},1,1)$ holds. Finally, the latter means that there exists a ladder system $\vec L=\langle A_\delta \mathrel {|} \delta \in S\rangle $ such that the following two hold:

1. (i) for every cofinal $A\subseteq \kappa $ , for every $n<\omega $ , there exists a $\delta \in S_n$ such that $\sup (A_\delta \cap A)=\delta $ ;

2. (ii) for every pair $\gamma <\delta $ of ordinals from S, $\sup (A_{\gamma }\cap A_{\delta })<\gamma $ .

Now letting $g:S\rightarrow \omega $ describe the partition of S, we get that for every function $f: \kappa \rightarrow \omega $ , by picking $n<\omega $ such that $A:=f^{-1}\{n\}$ is cofinal in $\kappa $ , we may find $\delta \in S_n$ such that $\sup (A_\delta \cap A)=\delta $ , and hence $\sup \{\alpha \in A_\delta \mathrel {|} f(\alpha ) =g(\delta )\} =\delta $ . So Clause (i) implies that $X_{\vec L}$ is not countably metacompact by Fact 4.2, and Clause (ii) ensures that $X_{\vec L}$ is a regular Hausdorff space by Fact 4.1.