Let $X$ be the Stone space of the free $\sigma$-algebra $A$ on $\omega\_1$ free generators.


Is $X$ separable (i.e. does $X$ contain a countable dense set)?



Separability of the Stone space of a free sigma-algebra
No.


Just looking at countably many generators we can produce a continuum of pairwise disjoint clopen subsets of $X$. Moreover, since $|A|=2^{\aleph\_0}$, we have that $2^{\aleph\_0} \leq c(X) \leq d(X) \leq w(X) \leq 2^{\aleph\_0}$, where $c$, $d$ and $w$ denote *cellularity*, *density* and *weight* respectively.


