My question is about existing of basis of club filter club($\omega\_1$) with cardinality $c$. Does it exist?



Club filter basis in $\omega_1$
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.


