# MA(countable), strong Choquet games and the perfect set property

Definition. A family $$\mathcal{F}\subseteq\mathcal{P}(\omega)$$ has the finite intersection property if $$\bigcap\sigma$$ is infinite for all $$\sigma\in[\mathcal{F}]^{<\omega}$$.

Definition. An $$A$$-triple to be a triple of the form $$(\mathcal{T},A,Q)$$ such that the following conditions are satisfied.

1) $$\mathcal{T}$$ is a strong Choquet, second-countable topology on $$2^{\omega}$$ that is finer than the standard topology (cones topology).

2) $$A\in\mathcal{T}$$.

3) $$Q$$ is a non-empty countable subset of $$A$$ with no isolated points in the subspace topology it inherits from $$\mathcal{T}$$.

I am reading the following Lemma and I found two details that I don't see very clearly.

1) Reviewing the proof, I don't understand how $$p'\in D_{\ell}^{ref}$$, precisely, I'm not sure how to show that the collection of open $$\{U_{s}^{p'}:s\in{^{n_{p'}}2}\}=\{U_{s}^{p'}:s\in{^{n_{p}+1}2}\}$$ it's a cover of $$2^{\omega}$$.

2) Again, in Lemma 30, my idea to prove that $$\mathcal{F}\cup\{\bigcap P\}$$ has the property of finite intersection is to assume otherwise. So, since $$\mathcal{F}$$ does have the property of finite intersection, there is $$\sigma=\{x_{1},...,x_{k}\}\in [\mathcal{F}]^{<\omega}$$ such that \begin{align*} H=\left(\displaystyle{\bigcap_{i = 1}^{k}x_{i}}\right)\cap\left(\bigcap P\right) \end{align*} then $$|H|<\omega$$. Since $$H$$ is finite, let $$\ell'=maxH$$. Let's take $$\ell=\ell'+1$$ and consider the dense set $$D_{\sigma,\ell}$$. Since $$G$$ is a $$\mathcal{D}$$-generic filter, there is $$p\in G\cap D_{\sigma,\ell}$$. If $$x=\bigcap P$$ then I must show that there is $$s\in{^{n_{p}}2}$$ such that $$x\in U_{s}^{p}$$ because there will be $$i>\ell$$ such that $$i\in\left(\displaystyle{\bigcap_{i = 1}^{k}x_{i}}\right)\cap\left(\bigcap P\right)$$ by the definition of $$D_{\sigma,\ell}$$ contradicting that $$\ell'$$ is the maximum of $$H$$. But I could not do it.

Any hint will help me a lot.

• A link to the paper quoted at the arXiv. – Pedro Sánchez Terraf Aug 24 '19 at 13:14
• @PedroSánchezTerraf arxiv.org/pdf/1108.2533.pdf – Fernando Mauricio Rivera Vega Aug 24 '19 at 15:55
• My previous comment is already linked. (If asking, I'd normally say "Please" :-).) Btw, I'm working on an answer now. The first part is ready; if you like, I can start by disclosing that. – Pedro Sánchez Terraf Aug 24 '19 at 16:19
• @PedroSánchezTerraf True, sorry, I didn't see the link. As you prefer. I don't know if you prefer to give a partial or total answer. Thank you very much for your help. By the way, the question I asked a few weeks ago about $\mathfrak{c}$ homeomorphisms I already have it thanks to the idea you wrote. At night I write it. – Fernando Mauricio Rivera Vega Aug 24 '19 at 22:38
• I'm delayed with the second part... I'll tell you here the first so you can go checking it (I confess I didn't do it thoroughly). I believe that they are only using “refines” in the sense of being just finer, not a finer partition. Perusing the second picture (p. 16) the density of $D_{\ell}^{\mathrm{ref}}$ seems to be only needed to show that $\bigcap_{n \in \omega} U_{x | n}$ is a singleton. – Pedro Sánchez Terraf Aug 24 '19 at 23:54

Concerning your first question, I believe that they are only using “refines” in the sense of being just finer, not a finer partition. Perusing the second picture (p. 16) the density of $$D_{\ell}^{\mathrm{ref}}$$ seems to be only needed to show that $$\bigcap_{n \in \omega} U_{x | n}$$ is a singleton.