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I am interested in the name for the following property of a Boolean algebra $\mathcal A$ of subsets of a set $X$:

$(\star)$ for any sequence $(A_n)_{n\in\omega}$ of pairwise disjoint nonempty sets in $\mathcal A$ there exists a subset $B\subseteq X$ such that the set $\{n\in\omega:A_n\cap B\ne\emptyset\}$ is infinite and the Boolean algebra $\{A\cap B:A\in \mathcal A\}$ is countable.

Is it OK to call Boolean algebras $\mathcal A$ with property $(\star)$ selectively countable?

Or there is some known and well-accepted name for $(\star)$?

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  • $\begingroup$ Topologically it seems to mean, in a Stone space: for every sequence of pairwise disjoint non-empty clopen subsets $(U_n)$, there exists a metrizable clopen subset $V$ such that $V\cap U_n$ is non-empty for all $n$. (Note that this already implies that the union of open metrizable subsets is dense, but conversely $\beta\omega$ doesn't satisfy this property, as any infinite $X$ in which no infinite clopen subset is metrizable.) $\endgroup$
    – YCor
    Commented Nov 22, 2021 at 10:24
  • $\begingroup$ @YCor Thank you for the comment, from which I understood and I posed a question too generally. So, I will rewrite the question to ask what I indeed wanted. $\endgroup$
    – Taras Banakh
    Commented Nov 22, 2021 at 10:33

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