Dense subset relation

Question

Defn Let $B$ be a Boolean algebra. A subset $D$ of $B$ is called b-dense if for every $0\neq b\in B$, there is $0\neq d\in D$ such that $d\leq b$.

Defn Let $T$ be a topological space. A subset $D$ of $T$ is called t-dense if the closure $Cl(D)=T$.

Let $T=Stone(B)$, the Stone space of $B$. I am asking what is the relation between b-dense in $B$ and t-dense in $T$.

Any idea ??

Answer 1

Dense subsets of a Boolean algebra correspond to $π$-bases consisting of clopen subsets of corresponding Stone space.

A collection of non-empty open subsets of a topological space is called $π$-base if every non-empty open set of the space contains some member of the collection as a subset.

$π$-bases are somewhere between bases and dense subsets. Each base is a $π$-base and union of $π$-base (or choosing a point in each set) is a dense set.