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.



Is there a calibre $\aleph_1$ Moore space which is not separable
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](https://zbmath.org/?q=an:0382.54003).)


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


