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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 $\bigcap_{\alpha\in\Lambda}U_\alpha\neq\emptyset$.

Is there a calibre $\aleph_1$ Moore space which is not separable?

(Under CH, the answer is no, since every first countable calibre $\aleph_1$ space is separable.)

Thanks.

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Frank Tall proved that under $MA+\neg CH$ there is a regular first-countable space with caliber $\omega_1$ which is not separable. (see Tall, Franklin D., First countable spaces with caliber $\aleph_1$ may or may not be separable, Set-theor. Topol., Vol. dedic. to M.K. Moore, 353-358 (1977). ZBL0382.54003.)

Applying the Moore Machine to Tall's example you get what you're looking for.

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    $\begingroup$ Does the Moore space generated by using Moore machine for $X$ keep property calibre $\aleph_1$ if $X$ is calibre $\aleph_1$? Could you provide a link or paper about this for me? $\endgroup$ – Paul Nov 12 '17 at 12:56

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