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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.

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  • $\begingroup$ A link to the paper quoted at the arXiv. $\endgroup$ – Pedro Sánchez Terraf Aug 24 '19 at 13:14
  • $\begingroup$ @PedroSánchezTerraf arxiv.org/pdf/1108.2533.pdf $\endgroup$ – Fernando Mauricio Rivera Vega Aug 24 '19 at 15:55
  • $\begingroup$ 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. $\endgroup$ – Pedro Sánchez Terraf Aug 24 '19 at 16:19
  • $\begingroup$ @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. $\endgroup$ – Fernando Mauricio Rivera Vega Aug 24 '19 at 22:38
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    $\begingroup$ 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. $\endgroup$ – Pedro Sánchez Terraf Aug 24 '19 at 23:54
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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.

I'll address the second question later.

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