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Suppose that $X$ is the space with point-countable base and $\aleph_1$ is a caliber of $X$. Must $X$ be second countable? Thanks for any help.

A topological space has calibre $\aleph_1$ if for every uncountable sequence $\langle U_\alpha\mid\alpha\lt\aleph_1\rangle$ of nonempty open sets $U_\alpha\subset X$, there is an uncountable subfamily $\Lambda\subset\aleph_1$ with $\bigcap_{\alpha\in\Lambda}U_\alpha\neq\emptyset$.

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Hint: If $X$ has $\omega_1$ as a calibre, then any point-countable family of nonempty open subsets of $X$ must be countable.

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