My question is about existing of basis of club filter club($\omega_1$) with cardinality $c$. Does it exist?
1 Answer
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That's independent of ZFC.
On the one hand, it's consistent with ZFC that $2^{\aleph_1}=\mathfrak c$, in which case the whole club filter on $\aleph_1$ has cardinality $\mathfrak c$.
On the other hand, the continuum hypothesis is consistent with ZFC and implies that the club filter has no basis of size $\mathfrak c=\aleph_1$. That's because the diagonal intersection of any $\aleph_1$ club sets is again club.