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Suppose that a topological space $X$ satisfies the following property

(P): "Each uncountable disjoint open family is locally countable",

where a family $\mathcal U$ of subsets of $X$ is called locally countable if each point of this space has a neighbourhood that intersects at most countably members of $\mathcal U$.

We do not know whether this property have been studied before.

If yes, what this property is called? Any reference/suggestion would be a great help for us.

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    $\begingroup$ Maybe this is just local countable cellularity (each point has a neighborhood with countable Souslin number)? $\endgroup$
    – Taras Banakh
    Commented Jun 28, 2022 at 18:01

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